Class Field Theory. Travis Dirle. December 4, 2016

Size: px
Start display at page:

Download "Class Field Theory. Travis Dirle. December 4, 2016"

Transcription

1 Class Field Theory

2 2

3 Class Field Theory Travis Dirle December 4, 2016

4 2

5 Contents 1 Global Class Field Theory Ray Class Groups The Idèlic Theory Artin Reciprocity The Existence Theorem and its Consequences Local Class Field Theory Preliminaries on Local Fields A Fundamental Exact Sequence Local Units Modulo Norms Lubin-Tate Extensions The Local Artin Map i

6 CONTENTS ii

7 Chapter 1 Global Class Field Theory 1.1 RAY CLASS GROUPS Definition If S is a set of prime ideals of O F, and lim s 1 + p S Np 1 log ( ) = δ exists, 1 s 1 then we say that S has Dirichlet density δ = δ F (S). Corollary Let K/F be Galois, and let Then δ F (S K/F ) = 1/[K : F ]. S K/F = {p O F : p splits completely in K/F }. Definition Let S, T be sets of primes in O F, where F is a number field. We define the following notation. Write S T to mean δ F (S\T ) = 0. Write S T if S T S. Theorem Let E and K be number fields, each of which is Galois over Q. Then S K S E if and only if E K. Theorem (Approximation Theorem) Let 1, n be non-trivial pairwise inequivalent absolute values on a number field F, and let β 1,..., β n be non-zero elements of F. For any ɛ > 0, there is an element α F such that α β j j < ɛ, for each j = 1,..., n. 1

8 Note that when p is a prime ideal of O F and c = π p for π p p 2, the statement αβ 0 and α β p < ɛ gives ord p ( α 1) > n, where n is given by β ɛ β p < c n. If α and β are p-adic units, then this just means α β mod p n. Note also that when j = σ, where σ : F R, the statement αβ 0 and α β σ < ɛ for small ɛ means that σ(α/β) > 0. In the p-adic situation, we were able to write α β mod p n. We can write something similar in the case of real embeddings σ of F if we make the following convention. Definition When σ : F R, we associate to σ a formal object that we call an infinite real prime, which we denote by p σ. We may then define α β mod p σ if and only if σ(α/β) > 0. We may also define infinite imaginary primes: We associate an object p σ to each conjugate pair σ, σ : F C. We dont use the congruence notation with infinite imaginary primes however. Definition Other language used for prime ideals can be adapted to infinite primes as well. If K/F is an extension of number fields, we say that an infinite prime p σ ramifies in K/F if and only if σ(f ) R, but for some extension of σ to K we have σ(k) R. Definition Using the infinite real primes (and the usual primes), we may also define a divisor or modulus for F as a formal product Π p p t(p), where t(p) N is non-zero for only finitely many p, and can only take a value of 0 or 1 when p is an infinite real prime. (We may consider the notion of an infinite imaginary prime, but if we do, we must take t(p) = 0 for all infinite imaginary primes p) Specifically, we shall denote the product of all the infinite real primes by m = Π σ real p σ. Definition If an element α F satisfies σ(α) > 0 for every real embedding σ of F, we say that α is totally positive, and write α 0. Definition Let m be a non-zero integral ideal of O F. Define P + F,m to be the subgroup of P F, which is { α : α O F, α 1 mod m, and α 0} or also, { α β : α β 0; α, β O F prime to m; α β mod m}. Definition Let I F (m) be the group of fractional ideals of F whose factorizations do not contain a non-trivial power of any prime ideal dividing m: 2 I F (m) = {a I F : ord p a = 0 for all p m}.

9 Definition The strict (narrow) ray class group or generalized ideal class group of F for m, is R + F,m = I F (m)/p + F,m. We write α 1 mod m when α 1 mod p ordp(m) in the completion F p for every p m. When writing congruences modulo powers of p in the completion, we really mean congruences modulo powers of the unique maximal ideal in the ring of integers of F p. Definition For a non-zero integral ideal m of O F modulo m as P F,m = { α : α 1 mod m}. Definition The ray class group of F for m is we define the ray R F,m = I F (m)/p F,m. The strict ray class group R + F,m may also be viewed as a ray class group in the above sense if one views P + F,m as a ray modulo the divisor mm. When m = O F, we have R F,m = I F /P F = C F, the ordinary ideal class group R + F,m = I F /P + F, the strict (narrow) ideal class group. Definition A generalized Dirichlet character or Weber character of modulus m is a homomorphism of groups χ : R + F,m C. Proposition R + F,m is a finite group, with where #R + F,m = h F 2 r 1 φ(m) [U F : U + F,m ] h F = #C F r 1 = # of real embeddings of F φ(m) = #(O F /m) = Π p m Np ep 1 (Np 1), where m = Π p m p ep U F = O F, the units of O F U + F,m = {ɛ U F : ɛ 0, ɛ 1 mod m} 3

10 Definition #R + F,m is called the strict ray class number modulo m or the ray class number modulo mm. Definition Let K/F be Galois, and let m be an integral ideal of O F. We say that K is the class field over F of P + F,m if S K/F = {primes p of O F : p splits completely in K/F } {primes p of O F : p P + F,m }. Recall, that S T if and only if they differ by a set with Dirichlet density zero. Definition More generally, we may define the notion of class field for subgroups of I F (m) that contain P + F,m. If m is a non-zero integral ideal of O F, and H satisfies P + F,m < H < I F (m), then we say K is the class field over F of H if K/F is Galois and S K/F {primes p of O F : p H}. Theorem If the class field K of H exists, then it is unique. Definition For a I F (m) we may define S a,m = {primes p of O F : p a in R + F,m } = {primes p ap+ F,m }. Proposition Let a I F (m). Suppose P + F,m < H < I F (m). If for all characters χ χ 0 of I F (m) that are trivial on H, we have L m (1, χ) 0, then δ F ({primes p of O F : p ah}) = 1 [I F (m) : H]. Theorem Suppose K/F is Galois, and P + F,m < H < I F (m). Suppose there is some set of primes J H with S K/F J. Then [I F (m) : H] [K : F ], and L m (1, χ) 0 whenever χ χ 0 and χ is trivial on H. Corollary If K/F is Galois and K is the class field for H where P + F,m < H < I F (m), then [I F (m) : H] = [K : F ]. Theorem (Universal Norm Index Inequality) Let K/F be a Galois extension of number fields and let H = P + F,m N K/F (m) where N K/F (m) = {a I F (m) : a = N K/F (A) for some A in I K }. (Note that the factorization of the fractional ideal A of K cannot contain a nontrivial power of any prime ideal that divides mo K, i.e., A I K (mo K ).) Then 4 [I F (m) : H] [K : F ].

11 It is possible to rephrase what we have done in terms of divisors. For a divisor m = Π p p a(p) of F, we shall write m 0 = Π finite p p a(p) and m re = Π real p p a(p). Of course, if p is real, then a(p) is either 0 or 1, and in general, we have a(p) = 0 for all but finitely many p. Given a divisor m of F, we write α 1 mod m to denote α 1 mod m 0 (i.e., ord p (α 1) ord p (m 0 ) for all p dividing m 0 ), and that σ(α) > 0 whenever σ is a real embedding with p σ dividing m re. Definition Remembering that m is a divisor of F (and not necessarily an ideal), we let P F,m denote the set of principal fractional ideals of F that have a generator α with α 1 mod m. (P F,m is sometimes called the ray modulo the divisor m). Also, set I F (m) = I F (m 0 ). We call R F,m = I F (m)/p F,m the ray class group modulo the divisor m. 1.2 THE IDÈLIC THEORY Definition We say that two absolute values are equivalent if they induce the same topology. Definition A place of F is an equivalence class of non-trivial absolute values on F. Denote the set of places of F by V F Theorem Each of the places of F falls into one of the following three categories: i) Places that contain one of the p-adic absolute values given by α p = Np ordp(α) for a non-zero prime ideal p of O F. These are the finite/non-archimedean/discrete places of F. ii) Places that contain one of the absolute values α σ = σ(α) R, for some real embedding σ : F R of F. These are the infinite/real-archimedean places of F. iii) Places that contain one of the absolute values α σ = σ(α) 2 C, for some σ : F C, an imaginary embedding of F. These are the infinite imaginary/imaginary-archimedean places of F. Note that two distinct non-zero prime ideals of O F cannot produce absolute values that are equivalent, so there is a distinct finite place for each non-zero 5

12 prime ideal of O F. Similarly, distinct real embeddings produce inequivalent absolute values. For the imaginary embeddings, each place contains the two (equivalent) absolute values corresponding to a conjugate pair of embeddings. But, if two imaginary embeddings of F are not conjugate, then they give rise to inequivalent absolute values. Thus, there is a single place for each conjugate pair of imaginary embeddings of F. For a number field F, there are a finite number of infinite places. Also, given x F, there can be only finitely many prime ideals p of O F for which x p 1. For a non-zero prime ideal p of O F, we let v p denote the place containing p. For an embedding σ : F C, we let v σ denote the place containing σ. Conversely, for a finite place v V F, we let p v denote the associated prime ideal of O F. To simplify notation, we write ord v instead of ord p. For v V F, we may complete F with respect to v. Denote the completion by F v. Note that if v is a finite place, then F v = F p for some p of O F. If v is an infinite real place, then F v = R, while if v is an infinite imaginary place, then F v = C. Definition An idèle of a number field F is an infinite vector a = (..., a v,...) v VF where each a v is an element of its corresponding F v, and where a v U v for all but finitely many v. Definition The idèles of F form a multiplicative group, denoted J F = Π v F v, being a restricted topological product. We let E F = Π v VF U v, which is a subgroup of J F. We may give E F the product topology, where each U v has its metric topology. Definition A topological group is a group G that is also a topological space, for which multiplication and inversion are continuous. Proposition J F is a locally compact topological group. Proposition The quotient group J F /E F is isomorphic to I F, the group of fractional ideals of F. Definition We may view α F as an idèle (..., ι v (α),...), where ι v : F F v is an embedding of F into its completion at v. This gives an embedding, called the diagonal embedding, ι : F J F, where ι(α) = (..., ι v (α),...). Usually it will do no harm to identify α and ι(α), and we shall often write F when we really mean ι(f ). If we define a map η : J F I F by a = (..., a v,...) a = Π v finite p ordv(av) v, we find that η(α) = Π v finite p ordv(α) v = αo F and η(f ) = P F. 6

13 Proposition For a number field F, we have J F /F E F = CF = I F /P F. A given place v of F will lie above either the infinite real place of Q or above a finite place of Q corresponding to a prime p of Z. Above the place, we choose the ι v from the set of embeddings F F v C, so that each infinite place of F is represented exactly once. Similarly, for the finite places v above p, we want to choose the ι v from the embeddings F F v C p so that each place of F above p is represented exactly once. Let denote the usual absolute value on C. For the infinite places and their embeddings, we have x v = ι v (x) d (where d = 1 if v is real and d = 2 if v is imaginary, i.e., d = [F v : R]). For a finite place v = v p, where p lies above p, the embedding ι v : F F v C p satisfies x v = ι v (x) d p, where d = [F v : Q p ] and p is the p-adic absolute value of C p, normalized so that p p = 1/p. Proposition Let m be a non-zero integral ideal of O F, and define J + F,m = {a J F : a v > 0 for all real v, and a v 1 Then E + F,m = J + F,m E F. J F /F E + F,m = R + F,m. mod p ordv(m) v for all p v m}, Corollary The set of subgroups H of J F, with H F E + F,m for some m, corresponds to the set of open subgroups of J F that contain F. Definition Let K/F be an extension of number fields. Define N K/F : J K J F as follows. Let (..., a w,...) = a J K, where the w are places of K. For a fixed v V F, the set {w V K : w v} is finite. We construct the norm of a as an idèle of F by computing each v-component in terms of the corresponding set {w V K : w v}. Specifically, we let b v = Π w v N Kw/Fv (a w ) and define N K/F (a) = (..., b v,...) J F Recall that if α K, then for any fixed v V F N K/F (α) = Π w v N Kw/F v (ι w (α)). Hence if α K is viewed as an idèle in J K, then N K/F (α) is the idèle in J F arising from the usual norm of the element α. Proposition Let K/F be abelian Galois, and let H = F N K/F J K (so F H J F ). Then H is an open subgroup in J F. Moreover, if m is chosen so that E + F,m H 7

14 then the image of H under the isomorphism J F /F E + F,m = I F (m)/p + F,m is precisely P + F,m N K/F (m)/p + F,m. We have [J F : H] = [I F (m) : P + F,m N K/F (m)] [K : F ]. Definition Let K/F be a (not necessarily abelian) Galois extension of number fields. We define an action of Gal(K/F ) on J K. Let G = Gal(K/F ) and let a = (..., a w,...) J K. Let σ G. For a place w of K, define the place σw by α σw = σ 1 (α) w or also σ(α) σw = α w. It is clear that G transitively permutes the places of K. σ induces an isomorphism between the completion that we also denote by σ : K w K σw. We may now define for each v V F σ(..., a w,...) w v = (..., b w,...) w v where b σw = σ(a w ), i.e., b w = σ(a σ 1 w) This gives an action of σ on J K. Since Π w v N Kw/F v (a w ) is the v th coordinate of N K/F (a), if we embed J F J K as before, we have Π σ G σ(a) = N K/F (a). Lemma Let C K = J K /K be the group of idèle classes of K, and similarly let C F = J F /F. The embedding J F J K induces an embedding C F C K. Furthermore, CK G = C F. Lemma Let k 2 /k 1 be an extension of local fields, (for some p, k j /Q p is a finite extension), with Gal(k 2 /k 1 ) = G, a cyclic group. Let U j denote the units of k j, i.e., the elements of absolute value 1. Then Q G (U 2 ) = 1, and [U G 2 : s(g)u 2 ] = [U 1 : N k2 /k 1 U 2 ] = e(k 2 /k 1 ) (whence also [ker s(g) : (σ 1)U 2 ] = e(k 2 /k 1 ). Lemma Let k 2 /k 1 be an extension of local fields and suppose we have subgroups B < A < U 2, with [A : B] = d. Then N k2 /k 1 B N k2 /k 1 A are subgroups of U 1 and [N k2 /k 1 A : N k2 /k 1 B] divides d. Corollary If k 2 /k 1 is an abelian extension of local fields, then 8 [U 1 : N k2 /k 1 U 2 ] e(k 2 /k 1 ).

15 Proposition For an abelian extension K/F of number fields with group G, let H = F N K/F J K. Then i) H is open in J F, so H E + F,m for some m, ii) the image of H under the isomorphism J F /F E + F,m = I F (m)/p + F,m is precisely P + F,m N K/F (m)/p + F,m. Proposition Let K/F be a Galois extension of number fields, with cyclic Galois group G = σ. Let v be a place of F and let w be a place of K above v. Then i) Q Gw (U w ) = 1 if w is finite, if w is real, or if v is imaginary. ii) Q Gw (U w ) = 2 if w is imaginary but v is real. iii) Q G (Π v S Π w v U w ) = 1, where S = {v V F : v is infinite, or v ramifies in K/F }. Lemma Let S be a finite set and let V = w S RX w be a real vector space. For an element w S a wx w of V, define a w X w 0 = max { a w : w S}, w w S (the sup-norm on V ). If {X w : w S} is given so that X w X w 0 < 1 dim R V for each w, then {X w : w S} is also a basis for V. Lemma Let G be a finite group acting on a finite set S. Let V = w S RX w be a vector space. Then G acts on V via ( ) σ a w X w = a w X σw. w S w S Note that the action of G preserves sup-norms: σ(x) 0 = X 0 for all X V. Let L V be a lattice preserved by G. Then there is a basis {Y w } w S of V contained in L such that σ(y w ) = Y σw for all σ G and for all w S. Proposition Let K/F be a Galois extension of number fields, with cyclic Galois group G = σ. Then Q G (U K ) = 2a [K : F ] where a = #{v V F : v is real on F, but extends to imaginary places on K}. Corollary Q G (C K ) = [K : F ]. 9

16 Theorem (Global Cyclic Norm Index Inequality) If K/F is a cyclic extension of number fields and m is an integral ideal of O F that is divisible by a sufficiently high power of every ramified prime in K/F, then [I F (m) : P + F,m N K/F (m)] = [K : F ]. 1.3 ARTIN RECIPROCITY Recall that for an ideal m of O F, we set J + F,m = {a J F : a v > 0 all real v, a v 1 mod p ordv(m) v all finite v}. E + F,m = J + F,m E F F m + = J + F,m F I F (m) = {a I F : ord p a = 0 for all p m} P + F,m = { α P F : α 0, α 1 mod m} N K/F (m) = {a I F (m) : a = N K/F (A) for some A I K } and showed J F /F E + F,m = J + F,m /E + F,m F + m = I F (m)/p + F,m via the homomorphism η m : J + F,m I F (m) given by a = (..., a v,...) a = Π v finite p ordv(av) v Definition There is a minimal ideal f of O F such that E + F,f H. This ideal f is called the conductor of H (or of K/F ), denoted f = f(k/f ). By minimality here, we mean that if E + F,m H then f m. By the minimality of f we have that if p v is unramified then p v f. The conductor cannot be divisible by any unramified prime. Definition Let K/F be a Galois extension of number fields with abelian Galois group G, and p a prime ideal of O F that is unramified in K/F. Then the decomposition group G( p = ) Z(p) must be cyclic (inertia is trivial) with a canonical generator σ p =, the Artin automorphism. 10 p K/F

17 Let m be an ideal of O F that is divisible by all the primes that ramify in the extension K/F and no others. The map p ( σ p ) induces a homomorphism A = A K/F : I F (m) G given by a σ a = where, for a = Π p p np I F (m), we set a K/F ( ) a σ a = Π p σ np p =. K/F (σ ( a does ) not depend on the choice of m). The map A is called the Artin map and is the Artin symbol. Note that since m is divisible by all the ramifying a K/F primes, σ p is defined for all p m. Proposition Let F L K, F E K be number fields and suppose K/F is abelian. Let p be a prime ideal of O F that is unramified in K/F and let P K be a prime ideal of O K that divides p. Let P L = P K L, P E = P K E (prime ideals of O E, O L, respectively, that divide p). Then ) ) f ( PE K/E L = ( p L/F where f = f(p E /p) is the residue field degree. Corollary Let F L K be number fields, where K/F is abelian Galois. Let p be a prime ideal of O F that is unramified in K/F. Then ( ) p ( ) p =. K/F L L/F Corollary Let F E K be number fields, where K/F is abelian Galois. Let p be a prime ideal of O F that is unramified in K/F and let P E be a prime of O E above p. Then ( ) PE = K/E ( ) NE/F P E. K/F Corollary Let K/F be an abelian Galois extension of number fields. Let m be an ideal of O F that is divisible by all the primes that ramify in K/F. Then N K/F (m) ker(a : I F (m) G). Theorem (Artin Reciprocity) Let K/F be an abelian extension of number fields, and assume m is an ideal of O F, divisible by all the ramifying primes. Let G = Gal(K/F ). Then i) A : I F (m) G is surjective, ii) the ideal m of O F can be chosen so that it is divisible only by the ramified primes and satisfies P + F,m ker(a); thus we have an epimorphism I F (m)/p + F,m G. iii) N K/F (m) ker(a). 11

18 Choosing m as in ii, we have a well defined homomorphism I F (m)/p + F,m N K/F (m) G Since the cardinality of the LHS is [K : F ] = #G by the Universal Norm Index Inequality, we have I F (m)/p + F,m N K/F (m) = G. Note that this isomorphism is given by the Artin map. Proposition If K/F is a cyclic extension of number fields with Galois group G, and m is an ideal of O F sufficiently large so that it is divisible by all the ramifying primes in K/F and so that E + F,m F N K/F J K, then the kernel of the Artin map satisfies ker(a) P + F,m N K/F (m). Lemma Ler r > 1, a > 1 be integers, and let q be a prime number. There is a prime number p such that the order of a mod p in (Z/pZ) is q r. Corollary Let a > 1 be an integer. Given q r as before, there are infinitely many primes p such that q r divides the order of a mod p in (Z/pZ). Lemma Let S be a finite set of primes, and let a > 1, n > 1 be integers. There is an integer d, prime to all the elements of S, such that n divides the order of a mod d in (Z/dZ). Lemma Given integers n > 1, a > 1, and a finite set S of primes, there is a positive integer m such that i) m is prime to all the elements of S ii) n divides the order of a mod m in (Z/mZ) iii) there exists b Z such that n divides the order of b mod m in (Z/mZ) but a and b are independent mod m (i.e., a mod m b mod m = 1). Lemma Let F be a number field, S a finite set of primes in Z, p a prime of O F. Then for any integer n > 1, there exists m Z, prime to S and to p, such that if ζ m is a primitive m th root of unity, then i) Gal(F (ζ m )/F ) = (Z/mZ). ( ii) p F (ζ m)/f ) has order divisible by n in Gal(F (ζ m )/F ). iii) there is some ( τ ) Gal(F (ζ m )/F ) of order divisible by n, such that τ is p independent to F (ζ m)/f. Note: independence implies τ Z(p) = 1, since ( ) generates the decomposition group Z(p). p F (ζ m)/f Lemma (Artin s Lemma) Let K/F be a cyclic extension of number fields of degree n, S a finite set of primes of Z, p a prime of O F. Then there is some m Z +, prime to the elements of S and to p, and an extension E/F such that i) K E = F 12

19 ii) K(ζ m ) = E(ζ m ), i.e., KE K(ζ m ) = E(ζ m ) iii) K F (ζ m ) = F iv) p splits completely in E/F. Suppose m satisfies Artin reciprocity and let S K/F = {prime ideals p of O F : p splits completely in K/F } J K/F = {prime ideals p of O F : p P + F,m N K/F (m)}. Corollary Let K/F be an abelian extension of number fields, say with [K : F ] = n, and let p be a prime of O F, unramified in K/F. Suppose m is divisible by all the ramified primes and no others, and suppose m satisfies Artin Reciprocity. Let f be the smallest positive integer such that p f P + F,m N K/F (m). Then, in O L, we have a factorization po L = P 1 P g, where each P i is a prime of O L with residue degree f over p, and where g = n/f. In particular, S K/F = J K/F. The statement of Artin Reciprocity reformulated in terms of idèles goes as follows. If m is chosen so that E + F,m F N K/F J K then we have J F J F /F N K/F J K I F (m)/p + F,m N K/F (m) Gal(K/F ) Definition Let ρ K/F : J F Gal(K/F ) be the composition. It is a surjective homomorphism of groups with kernel F N K/F J K. We say K is the class field over F of F N K/F J K and we call ρ K/F the idèlic Artin map. For a J F, we sometimes denote ( ) a ρ K/F (a) =. K/F 1.4 THE EXISTENCE THEOREM AND ITS CONSEQUENCES 13

20 Theorem (The Existence Theorem) Every open subgroup H J F with H F is of the form H = F N K/F J K for some (unique) finite abelian extension K/F Theorem (The Existence Theorem) For any H, with P + F,m < H < I F (m), there is a class field K/F associated to H. Theorem (The Completeness Theorem) For any abelian extension K/F, there is some m and some H with P + F,m < H < I F (m) such that K is the class field over F of H. Theorem (The Isomorphy Theorem) When P + F,m < H < I F (m), and K is the class field over F of H, we have Gal(K/F ) = I F (m)/h with the isomorphism being induced by the Artin map. Proposition Let Φ : {finite abelian extensions K of F} {open subgroups H of J F that contain F } be given by Φ(K) = F N K/F J K. Then: i) K K if and only if Φ(K ) Φ(K), the Ordering Theorem, ii) Φ(KK ) = Φ(K) Φ(K ), iii) Φ(K K ) = Φ(K)Φ(K ), iv) If H = Φ(E) = F N E/F J E and K E, then E is the fixed field of ρ K/F (H). Corollary Suppose K is the class field to the open subgroup H of J F, where H contains F, and let H H be an open subgroup of J F. Then H has a class field over F. Proposition (Reduction Lemma) Let K/F be a cyclic extension of number fields and suppose H is an open subgroup of J F that contains F. Let H K = {x J K : N K/F (x) H} = N 1 K/F (H). If H K has a class field over K, then H has a class field over F Definition An abelian extension K/F is said to have exponent n if the abelian group Gal(K/F ) has exponent n. Definition A finite abelian extension K/F is called a Kummer n-extension if Gal(K/F ) is a group with exponent n and F contains all the n th roots of unity. Theorem Let F be a number field containing all the n th roots of unity. There is a bijective correspondence between the finite Kummer n-extensions K of F and the subgroups W of F with (F ) n W and W/(F ) n finite. The correspondence associates W to the field K = F (W 1/n ), for which we have Gal(K/F ) = W/(F ) n. 14

21 Let F be a number field. Let S be a finite set of places of F and assume S S = {infinite places of F}. Define J F,S = Π v S F v Π v S U v an open subgroup of J F. F S = J F,S F the S-units of F, a discrete subgroup of J F,S. Note that F S also may be defined without using idèles: F S = {α F : the factorization of α involves no prime p v with v S}. Lemma There is a finite set of places S S such that J F = F J F,S. Theorem Let S be a finite set of places of F, with S S. Then F S is the direct product of the (finite cyclic) group of roots of unity in F, and a free abelian group of rank #S 1. That is F S = WF Z #S 1. Corollary If F contains the n th roots of unity and S S is a finite set of places of F, then [F S : F n S ] = n #S. Lemma Let v be a finite place of F, and let n Z +. If µ n denotes the set of all n th roots of unity, then i) [U v : Uv n ] = 1 n v #(F v µ n ). ii) [F v : (F v ) n ] = n n v #(F v µ n ). Theorem Let F be a number field that contains all the n th roots of unity. Let S be a finite set of places of F containing S, the places v such that p v n and sufficiently many finite places so that J F = F J F,S. Let B = Π v S (F v ) n Π v S U v. Then F B has class field F (F 1/n S ) over F Theorem (The Existence Theorem) Let F be a number field. Let H be an open subgroup of J F with F H. Then H has a class field over F, i.e., there is a finite abelian extension K of F such that H = F N K/F J K. Theorem Let H be an open subgroup of J F that contains F and let K be the class field of H over F. Let v be a place of F. If p v splits completely in K/F, then φ v (F v ) H, where φ is the embedding φ v : F v J F. Theorem Let H be an open subgroup of J F that contains F and let K be the class field over F of H. Suppose J F /H has exponent n and that F contains the n th roots of unity. Let v 0 be a place of F with φ v0 (F v 0 ) H. Then p v0 splits completely in K/F. 15

22 Theorem Let K/F be an abelian extension of number fields, and let v be a finite place of F. The Artin map ρ K/F satisfies ρ K/F (φ v (F v )) = Z(p v ), the decomposition group. Theorem (Complete Splitting Theorem) Let H be an open subgroup J F with F H, and let K be the class field of H over F. Let v be a finite place of F. Then p v splits completely in K/F if and only if φ v (F v ) H. Corollary Let H be an open subgroup of J K with F H and let K be the class field of H over F. Then φ v (F v ) H = φ v (N Kw/F v K w ) φ v (U v ) H = φ v (N Kw/F v U w ). Theorem Let K/F be an abelian extension of number fields, and let H = F N K/F J K. Let v be a finite place of F. Then ρ K/F (φ v (U v )) = T (p v ), the inertia subgroup in Gal(K/F ). Corollary If H is an open subgroup of J F with F H, and K is the class field to H over F, then for any finite place v of F, the class field to Hφ v (U v ) is the maximal subfield of K in which p v is unramified, hence it is the field K T, (the fixed field of T (p v )). Corollary Let K/F be an abelian extension of number fields, v a finite place of F, and w a place of K above v. Then U v /N Kw/F v U w = T (pv ). Theorem If K/F is an abelian extension of number fields, then f(k/f ) is divisible by all the ramified primes, and no others. Theorem (Kronecker, Weber) Every finite abelian extension F of Q satisfies F Q(ζ) for some root of unity ζ. Proposition Let F be a number field and let H be an open subgroup of J F that contains F. Then H F E F if and only if the class field to H over F is an abelian extension of F that is everywhere unramified. Definition Taking H = F E F and applying previous proposition, we find that the extension F 1 /F (where F 1 is the class field of H) is abelian and everywhere unramified; it is necessarily the maximal unramified abelian extension of F. F 1 is called the Hilbert class field of F. Theorem (Principal Ideal Theorem) Every fractional ideal a of a number field F becomes principal in F 1, i.e., ao F1 is principal. Theorem Let K/F be an extension of number fields and suppose K F 1 = F. Then i) h F h K. ii) the map N K/F : C K C F is surjective. 16

23 Proposition If K/F is an extension of number fields, and there is some prime p of O F that is totally ramified in K/F, then h F h K. Theorem Let E/F be an extension of number fields and let H = F N E/F J E, an open subgroup of J F that contains F. Let K be the class field of H over F. Then K/F is the maximal abelian subextension of E/F. Let E/F be an arbitrary (not necessarily Galois) extension of number fields and put S 1 E/F = {unramified primes p of O F : f(p/p) = 1 for some prime P po E }. Theorem Suppose K/F is a Galois extension of number fields and L/F is any finite extension. Then S 1 L/F S K/F if and only if K L. Corollary A Galois extension of number fields K/F is uniquely determined by the set S K/F of primes that split completely. Definition Let M/F be a (possibly infinite) Galois extension with Galois group G. For each σ G, we take the cosets {σgal(m/k) : K/F is a finite subextension of M/F } as a basis of open neighborhoods of σ. The resulting topology is called the Krull topology on G. Theorem (Main Theorem of Galois Theory - General Case) Let M/F be a Galois extension with Galois group G. The map L Gal(M/L) is a bijective correspondence between the subextension L/F of M/F and the closed subgroups of Gal(M/F ). Moreover, in this correspondence the open subgroups of Gal(M/F ) are paired with the finite subextensions of M/F. Theorem Let p > 2 be a prime, let E = Q(ζ p ) have class number h, and let E + = Q(ζ p + ζp 1 ) have class number h +. Let h = h/h +. Then h 1 = 2pΠ χ odd,fχ=p L(0, χ) 2 h + = [U E : Y E ] where Y E is the group of cyclotomic units of E, i.e., { 1 ζ a } p Y E = : a, b 0 mod p. 1 ζp b Theorem Let p > 2 be a prime, let E = Q(ζ p ), and let E + = Q(ζ p + ) as before. The map C E + C E given by [a] E + [ao E ] E is injective. ζ 1 p 17

24 Theorem Let E = Q(ζ p ) and let B n denote the n th Bernoulli number, i.e., t e t 1 = t n B n n!. Then p h E if and only if p divides the numerator of some B 2k, where 1 k < p 1 2. Theorem If p > 2 is prime and p h E, where E = Q(ζ p ), then has no non-trivial solution in integers. n=0 x p + y p = z p, (xyz, p) = 1 Lemma If K/E is everywhere unramified and Galois with Gal(K/E) = G, then I G K = I E. Theorem (Kummer) Let E = Q(ζ p ) where p > 2 is prime. If p h +, then p h. Proposition Suppose K/F is a Z p -extension, where F is a number field. Let q be a prime of F that does not divide po F. Then K/F is unramified at q. Proposition Let K/F be a Z p -extension, where F is a number field. Some prime of O F ramifies in K/F and moreover there is some level m such that every prime that ramifies in K/K (m) is totally ramified. 18

25 Chapter 2 Local Class Field Theory 2.1 PRELIMINARIES ON LOCAL FIELDS Here we let K be a field that is complete with respect to a normalized discrete valuation v K : K Z. We could just assume that K is an extension field of Q p (of possibly infinite degree). Let We also denote the following O K = {x K : v K (x) 0} U K = {x K : v K (x) = 0} π K, a uniformizer in K, so v K (π K ) = 1 P K = {x K : v K (x) > 0} = π K O K U m K = {x U K : x 1 mod P m K } F K = O K /P K, the residue field of K K ur a maximal unramified extension of K K ur the completeion of K ur F ur the residue field of K ur Ω a complete, algebraically closed extension of K ur Consider the case when K is local. Let K (t) denote the unramified extension of K of degree t, i.e., the splitting field over K of the polynomial X qt X, where q = #F K. Clearly K (t) K (n) if and only if t n. The field K ur is simply the union K (t). 19

26 CHAPTER 2. LOCAL CLASS FIELD THEORY Proposition Let K be a local field. The field F ur is an algebraic closure of F K. Moreover, there is a natural isomorphism Gal(K ur /K) = Gal(F Kur /F K ). Definition The Frobenius automorphism is just the lift to Gal(L/K) of the map x x q from Gal(F L /F K ), where q is the cardinality of the residue field of K. But the map x x q also can be viewed as belonging to Gal(F Kur /F K ); its life to Gal(K ur /K) will be called the Frobenius automorphism of K, denoted φ. Proposition Let K be a local field and suppose L/K is Galois with K ur L. Let σ Gal(L/K) be such that σ Kur = φ. Let F be the fixed field of σ in L. Then F K ur = L and F K ur = K, so Gal(L/F ) = Gal(K ur /K). Note this implies F/K is totally ramified. Theorem (Decomposition Theorem) Let L/K be a finite Galois extension, and suppose F K is a finite field. There is a totally ramified extension L /K such that L ur = L K ur = LK ur = L ur. Moreover, if Gal(L/K) ram is central in Gal(L/K), then we can take L /K to be abelian. 2.2 A FUNDAMENTAL EXACT SEQUENCE Lemma Let G be a finite abelian group and let g G. Then G contains a subgroup H such that G/H is cyclic and the order of gh in G/H is the same as the order of g in G. Throughout this section, we suppose K is complete with respect to a discrete valuation and F K is algebraically closed. For the extension L/K, we put a subgroup of U L. V(L/K) = σ(u) u : u U L, σ Gal(L/K), Proposition Let L/K be a finite abelian extension. The group homomorphism is injective. i : Gal(L/K) U L /V(L/K) given by i(σ) = σ(π L) V(L/K) π L Theorem Suppose F K is algebraically closed, and L/K is a finite cyclic extension. Let N : U L /V(L/K) U K be the map that sends the coset uv(l/k) 20

27 CHAPTER 2. LOCAL CLASS FIELD THEORY to N L/K (u). Since elements of V(L/K) have norm 1, this is a well-defined homomorphism. The sequence is exact. 1 Gal(L/K) i U L /V(L/K) N U K 1 Lemma Let L/K be a finite Galois extension, and let K E L, where E/K is Galois. Then N L/E V(L/K) = V(E/K). Lemma Let L/K be a finite abelian extension, let E be an intermediate field such that L/E is cyclic. Then 1 Gal(L/E) i U L /V(L/K) Ñ U E /V(E/K) 1 is exact, where the map Ñ is induced by N L/E. Theorem If F K is algebraically closed and L/K is a finite abelian extension, then is an exact sequence. 1 Gal(L/K) i U L /V(L/K) N U K LOCAL UNITS MODULO NORMS Throughout this section, we suppose K is a local field (so F K is finite). Let L/K be a finite abelian extension that is totally ramified. Then the extension L ur / K ur is abelian and totally ramified with Gal( L ur / K ur ) = Gal(L/K). Recall we let F ur denote the residue field of K ur (and of K ur ), so that F ur is an algebraic closure of F K. We use φ to denote the Frobenius automorphism in Gal(F ur /F K ); also we continue to use φ to denote its lifts in Gal(K ur /K) and Gal(L ur /L), and their extensions to K ur and L ur. We have a homomorphism φ 1 : U Kur U Kur given by u φ(u)u 1. Lemma We have the following: i) φ 1 : U Kur U Kur is surjective, as is φ 1 : O Kur O Kur, ii) φ 1 : V( L ur / K ur ) V( L ur / K ur ) is surjective, iii) ker(φ 1 : U Kur U Kur ) = U K. 21

28 CHAPTER 2. LOCAL CLASS FIELD THEORY Proposition Define a map θ L/K : U K Gal(L/K) for abelian extensions L/K that are totally ramified. Then i) θ L/K is surjective ii) kerθ L/K = N L/K U L. Theorem For any finite totally ramified abelian extension L/K, there is an isomorphism θ L/K : U K /N L/K U L Gal(L/K). Theorem If L/K is a finite abelian extension, then there is a canonical isomorphism θ L/K : U K /N L/K U L Gal(L/K) ram. 2.4 LUBIN-TATE EXTENSIONS Definition For two formal power series F, G we write F G mod deg d to mean that F and G coincide in terms of degree less than d. Definition A one-dimensional formal group law over R is a power series, F R [[X, Y ]] such that i) F (X, 0) = X, F (0, Y ) = Y, and ii) F (X, F (Y, Z)) = F (F (X, Y ), Z). If we also have F (X, Y ) = F (Y, X), then F is said to be a commutative formal group law. Definition Suppose F and G are one-dimensional formal group laws over R. A power series θ R [[X]] that satisfies i) θ(x) 0 mod deg 1, and ii) θ(f (X, Y )) = G(θ(X), θ(y )) is called an R-homomorphism from F to G. Denote the set of all R-homomorphisms from F to G by Hom R (F, G). Let K be a local field. For each uniformizer π of K, let F π = {f(x) O K [[X]] : f(x) πx mod deg 2 and f(x) X q mod P K } Theorem (Lubin, Tate) Let π be a uniformizer in a local field K, and let F K have order q. Suppose f, g F π and let l(x 1,..., X m ) = a 1 X

29 CHAPTER 2. LOCAL CLASS FIELD THEORY a m X m be a linear form (with a i O K ). Then there is a unique power series F O K [[X 1,..., X m ]] such that F (X 1,..., X m ) l(x 1,..., X m ) mod deg 2 f(f (X 1,..., X m )) = F (g(x 1 ),..., g(x m )). Definition For f F π, let F f (X, Y ) be the unique power series in O K [[X, Y ]] that satisfies F f (X, Y ) X + Y mod deg 2, f(f f (X, Y )) = F f (f(x), f(y )). The formal group laws F f for f F πk are called the Lubin-Tate formal group laws for π K. Lemma (Lubin, Tate) Suppose K is a local field, with residue field F K of order q. Let π be a uniformizer in K, and let f(x), g(x) F π. Then for any a O K there is a unique power series [a] f,g (X) O K [[X]] such that f([a] f,g (X)) = [a] f,g (g(x)) [a] f,g (X) ax mod X 2 Corollary (Lubin, Tate) Let K be a local field and let π be a uniformizer in K. Suppose f(x), g(x), h(x) F π and a, b O K. As is customary, we put [a] f = [a] f,f. i) [π] f (X) = f(x) ii) [a] f,g ([b] g,h (X)) = [ab] f,h (X) for any a, b O K iii) [1] f,g ([1] g,f (X)) = X iv) [a] f,g (F g (X, Y )) = F f ([a] f,g (X), [a] f,g (Y )) v) [a + b] f,g (X) = F f ([a] f,g (X), [b] f,g (X)). Lemma Let k be any field, and let g(x) = X n + a n 1 X n a 0 k[x] where either char k = 0 or n is prime to char k. Then we may find a positive integer r and a polynomial g(x) k[x] of degree less than r, such that the polynomial h(x) = X r g(x) + g(x) has only simple zeros. Proposition Let K be a local field and fix a polynomial f(x) F πk. For m Z +, let L m be the field associated to f(x). Then N Lm/KU Lm U m K. Lemma Let f(x) O K [[X]], and suppose L/K is a finite extension. If there is some λ L with v L (λ) > 0 and f(λ) = 0, then there is a power series h(x) O K [[X]] with f(x) = (X λ)h(x). Lemma Suppose u, u U K and let f(x) F πk be a polynomial of degree q. If [u] f (λ m ) = [u ] f (λ m ) then uuk m = u UK m. 23

30 CHAPTER 2. LOCAL CLASS FIELD THEORY Theorem Let K be a local field and let L m be as above, for some polynomial f(x) F πk of degree q. Then L m /K is Galois, and Gal(L m /K) = U K /U m K. Corollary The extensions L m /K depend only on the choice of uniformizer π K ; they do not depend on the choice of polynomial f(x) F πk. Definition The field L m is called the mth Lubin-Tate extension of K associated to the uniformizer π K. Corollary Let L M be the mth Lubin-Tate extension of a local field K. Then N Lm/KU Lm = U m K. Theorem Let K be a local field. There are isomorphisms Gal(K ab /K) ram = U K and Gal(K ab /K) = U K Ẑ. Corollary Let K be a local field with uniformizer π K. Put L πk = L m where the L m are the Lubin-Tate extensions of K associated to the uniformizer π K. Then K ab = L πk K ur. 2.5 THE LOCAL ARTIN MAP Lemma If L/K is a finite abelian extension, then [K : N L/K L ] = [L : K]. Let ρ LK(t) /K : K Gal(LK (t) /K) be the unique homomorphism that satisfies ρ LK(t) /K(u) = θ L/K (u 1 ) for u U K ρ LK(t) /K(π K ) = φ Frobenius in Gal(LK (t) /L). Lemma Let K (t) /K be a finite unramified extension of K of degree t. Let L/K and E/K be finite totally ramified abelian extension of K such that LK (t) = EK (t). Consider the composition K ρ LK (t) /K Gal(LK (t) /K) nat Gal(E/K). The kernel of this composition is N E/K E. 24

31 CHAPTER 2. LOCAL CLASS FIELD THEORY Definition Choose a uniformizer π K of K, and use the union L πk of the Lubin-Tate extensions of K, recalling that K ab = L πk K ur, and we may identify Gal(K ab /K ur ) = Gal(L πk /K). Define ρ K : K Gal(K ab /K) to be the unique homomorphism that satisfies ρ K (u) = θ Kab /K(u 1 ) Gal(L πk /K) for u U K ρ K (π K ) = φ Frobenius in Gal(K ab /L πk ). Since π K N Lm/KL m for all the Lubin-Tate extensions L m /K, it follows that ρ K agrees with ρ LmK (t) /K. ρ K is called the local Artin map or the local norm residue map. Theorem Let L/K be a finite abelian extension. Consider the composition K ρ K Gal(K ab /K) restr Gal(L/K). The kernel of this composition is N L/K L. Corollary The open subgroups of finite index in K are precisely the subgroups of the form N L/K L, for L/K finite abelian. Indeed, any open subgroup of finite index in K is the kernel of the composition for some finite abelian extension L/K. K ρ K Gal(K ab /K) nat Gal(L/K) Lemma Let π and π be uniformizers in a local field K, say with π = uπ, where u U K. Let q = #F K and suppose f(x), g(x) are polynomials of degree q such that f(x) F π and g(x) F π. We use φ to denote the Frobenius automorphism in Gal(K ur /K) and also its extension to ˆK ur. Theorem Define the homomorphism γ π : K Gal(L π K ur /K), with γ π (u) = σ u 1 in Gal(L π K ur /K ur ) = Gal(L π /K), γ π (π) = φ Frobenius in Gal(L π K ur /L π ). γ π does not depend on the choice of uniformizer π. Moreover, γ π is the local Artin map, i.e., γ π = ρ K. 25

32 CHAPTER 2. LOCAL CLASS FIELD THEORY 26

FIELD THEORY. Contents

FIELD THEORY. Contents FIELD THEORY MATH 552 Contents 1. Algebraic Extensions 1 1.1. Finite and Algebraic Extensions 1 1.2. Algebraic Closure 5 1.3. Splitting Fields 7 1.4. Separable Extensions 8 1.5. Inseparable Extensions

More information

CLASS FIELD THEORY WEEK Motivation

CLASS FIELD THEORY WEEK Motivation CLASS FIELD THEORY WEEK 1 JAVIER FRESÁN 1. Motivation In a 1640 letter to Mersenne, Fermat proved the following: Theorem 1.1 (Fermat). A prime number p distinct from 2 is a sum of two squares if and only

More information

A BRIEF INTRODUCTION TO LOCAL FIELDS

A BRIEF INTRODUCTION TO LOCAL FIELDS A BRIEF INTRODUCTION TO LOCAL FIELDS TOM WESTON The purpose of these notes is to give a survey of the basic Galois theory of local fields and number fields. We cover much of the same material as [2, Chapters

More information

Some algebraic number theory and the reciprocity map

Some algebraic number theory and the reciprocity map Some algebraic number theory and the reciprocity map Ervin Thiagalingam September 28, 2015 Motivation In Weinstein s paper, the main problem is to find a rule (reciprocity law) for when an irreducible

More information

A SIMPLE PROOF OF KRONECKER-WEBER THEOREM. 1. Introduction. The main theorem that we are going to prove in this paper is the following: Q ab = Q(ζ n )

A SIMPLE PROOF OF KRONECKER-WEBER THEOREM. 1. Introduction. The main theorem that we are going to prove in this paper is the following: Q ab = Q(ζ n ) A SIMPLE PROOF OF KRONECKER-WEBER THEOREM NIZAMEDDIN H. ORDULU 1. Introduction The main theorem that we are going to prove in this paper is the following: Theorem 1.1. Kronecker-Weber Theorem Let K/Q be

More information

The Kronecker-Weber Theorem

The Kronecker-Weber Theorem The Kronecker-Weber Theorem November 30, 2007 Let us begin with the local statement. Theorem 1 Let K/Q p be an abelian extension. Then K is contained in a cyclotomic extension of Q p. Proof: We give the

More information

Class Field Theory. Peter Stevenhagen. 1. Class Field Theory for Q

Class Field Theory. Peter Stevenhagen. 1. Class Field Theory for Q Class Field Theory Peter Stevenhagen Class field theory is the study of extensions Q K L K ab K = Q, where L/K is a finite abelian extension with Galois grou G. 1. Class Field Theory for Q First we discuss

More information

Profinite Groups. Hendrik Lenstra. 1. Introduction

Profinite Groups. Hendrik Lenstra. 1. Introduction Profinite Groups Hendrik Lenstra 1. Introduction We begin informally with a motivation, relating profinite groups to the p-adic numbers. Let p be a prime number, and let Z p denote the ring of p-adic integers,

More information

ON GALOIS GROUPS OF ABELIAN EXTENSIONS OVER MAXIMAL CYCLOTOMIC FIELDS. Mamoru Asada. Introduction

ON GALOIS GROUPS OF ABELIAN EXTENSIONS OVER MAXIMAL CYCLOTOMIC FIELDS. Mamoru Asada. Introduction ON GALOIS GROUPS OF ABELIAN ETENSIONS OVER MAIMAL CYCLOTOMIC FIELDS Mamoru Asada Introduction Let k 0 be a finite algebraic number field in a fixed algebraic closure Ω and ζ n denote a primitive n-th root

More information

Class Field Theory. Anna Haensch. Spring 2012

Class Field Theory. Anna Haensch. Spring 2012 Class Field Theory Anna Haensch Spring 202 These are my own notes put together from a reading of Class Field Theory by N. Childress [], along with other references, [2], [4], and [6]. Goals and Origins

More information

p-adic fields Chapter 7

p-adic fields Chapter 7 Chapter 7 p-adic fields In this chapter, we study completions of number fields, and their ramification (in particular in the Galois case). We then look at extensions of the p-adic numbers Q p and classify

More information

Local Class Field Theory I

Local Class Field Theory I CHAPTER 4 Local Class Field Theory I In this chapter we develop the theory of abelian extensions of a local field with finite residue field. The main theorem establishes a correspondence between abelian

More information

Math 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille

Math 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille Math 429/581 (Advanced) Group Theory Summary of Definitions, Examples, and Theorems by Stefan Gille 1 2 0. Group Operations 0.1. Definition. Let G be a group and X a set. A (left) operation of G on X is

More information

Galois theory of fields

Galois theory of fields 1 Galois theory of fields This first chapter is both a concise introduction to Galois theory and a warmup for the more advanced theories to follow. We begin with a brisk but reasonably complete account

More information

Part II Galois Theory

Part II Galois Theory Part II Galois Theory Theorems Based on lectures by C. Birkar Notes taken by Dexter Chua Michaelmas 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after

More information

HONDA-TATE THEOREM FOR ELLIPTIC CURVES

HONDA-TATE THEOREM FOR ELLIPTIC CURVES HONDA-TATE THEOREM FOR ELLIPTIC CURVES MIHRAN PAPIKIAN 1. Introduction These are the notes from a reading seminar for graduate students that I organised at Penn State during the 2011-12 academic year.

More information

1 Rings 1 RINGS 1. Theorem 1.1 (Substitution Principle). Let ϕ : R R be a ring homomorphism

1 Rings 1 RINGS 1. Theorem 1.1 (Substitution Principle). Let ϕ : R R be a ring homomorphism 1 RINGS 1 1 Rings Theorem 1.1 (Substitution Principle). Let ϕ : R R be a ring homomorphism (a) Given an element α R there is a unique homomorphism Φ : R[x] R which agrees with the map ϕ on constant polynomials

More information

The Kronecker-Weber Theorem

The Kronecker-Weber Theorem The Kronecker-Weber Theorem Lucas Culler Introduction The Kronecker-Weber theorem is one of the earliest known results in class field theory. It says: Theorem. (Kronecker-Weber-Hilbert) Every abelian extension

More information

Class Field Theory. Abstract

Class Field Theory. Abstract Class Field Theory Abstract These notes are based on a course in class field theory given by Freydoon Shahidi at Purdue University in the fall of 2014. The notes were typed by graduate students Daniel

More information

GALOIS EXTENSIONS ZIJIAN YAO

GALOIS EXTENSIONS ZIJIAN YAO GALOIS EXTENSIONS ZIJIAN YAO This is a basic treatment on infinite Galois extensions, here we give necessary backgrounds for Krull topology, and describe the infinite Galois correspondence, namely, subextensions

More information

The Kummer Pairing. Alexander J. Barrios Purdue University. 12 September 2013

The Kummer Pairing. Alexander J. Barrios Purdue University. 12 September 2013 The Kummer Pairing Alexander J. Barrios Purdue University 12 September 2013 Preliminaries Theorem 1 (Artin. Let ψ 1, ψ 2,..., ψ n be distinct group homomorphisms from a group G into K, where K is a field.

More information

Representation of prime numbers by quadratic forms

Representation of prime numbers by quadratic forms Representation of prime numbers by quadratic forms Bachelor thesis in Mathematics by Simon Hasenfratz Supervisor: Prof. R. Pink ETH Zurich Summer term 2008 Introduction One of the most famous theorems

More information

Math 121 Homework 6 Solutions

Math 121 Homework 6 Solutions Math 11 Homework 6 Solutions Problem 14. # 17. Let K/F be any finite extension and let α K. Let L be a Galois extension of F containing K and let H Gal(L/F ) be the subgroup corresponding to K. Define

More information

9 Artin representations

9 Artin representations 9 Artin representations Let K be a global field. We have enough for G ab K. Now we fix a separable closure Ksep and G K := Gal(K sep /K), which can have many nonabelian simple quotients. An Artin representation

More information

Field Theory Qual Review

Field Theory Qual Review Field Theory Qual Review Robert Won Prof. Rogalski 1 (Some) qual problems ˆ (Fall 2007, 5) Let F be a field of characteristic p and f F [x] a polynomial f(x) = i f ix i. Give necessary and sufficient conditions

More information

Contradiction. Theorem 1.9. (Artin) Let G be a finite group of automorphisms of E and F = E G the fixed field of G. Then [E : F ] G.

Contradiction. Theorem 1.9. (Artin) Let G be a finite group of automorphisms of E and F = E G the fixed field of G. Then [E : F ] G. 1. Galois Theory 1.1. A homomorphism of fields F F is simply a homomorphism of rings. Such a homomorphism is always injective, because its kernel is a proper ideal (it doesnt contain 1), which must therefore

More information

Galois theory (Part II)( ) Example Sheet 1

Galois theory (Part II)( ) Example Sheet 1 Galois theory (Part II)(2015 2016) Example Sheet 1 c.birkar@dpmms.cam.ac.uk (1) Find the minimal polynomial of 2 + 3 over Q. (2) Let K L be a finite field extension such that [L : K] is prime. Show that

More information

Algebraic Number Theory Notes: Local Fields

Algebraic Number Theory Notes: Local Fields Algebraic Number Theory Notes: Local Fields Sam Mundy These notes are meant to serve as quick introduction to local fields, in a way which does not pass through general global fields. Here all topological

More information

Ramification Theory. 3.1 Discriminant. Chapter 3

Ramification Theory. 3.1 Discriminant. Chapter 3 Chapter 3 Ramification Theory This chapter introduces ramification theory, which roughly speaking asks the following question: if one takes a prime (ideal) p in the ring of integers O K of a number field

More information

Fields. Victoria Noquez. March 19, 2009

Fields. Victoria Noquez. March 19, 2009 Fields Victoria Noquez March 19, 2009 5.1 Basics Definition 1. A field K is a commutative non-zero ring (0 1) such that any x K, x 0, has a unique inverse x 1 such that xx 1 = x 1 x = 1. Definition 2.

More information

GALOIS THEORY AT WORK: CONCRETE EXAMPLES

GALOIS THEORY AT WORK: CONCRETE EXAMPLES GALOIS THEORY AT WORK: CONCRETE EXAMPLES KEITH CONRAD 1. Examples Example 1.1. The field extension Q(, 3)/Q is Galois of degree 4, so its Galois group has order 4. The elements of the Galois group are

More information

24 Artin reciprocity in the unramified case

24 Artin reciprocity in the unramified case 18.785 Number theory I Fall 2017 ecture #24 11/29/2017 24 Artin reciprocity in the unramified case et be an abelian extension of number fields. In ecture 22 we defined the norm group T m := N (I m )R m

More information

Graduate Preliminary Examination

Graduate Preliminary Examination Graduate Preliminary Examination Algebra II 18.2.2005: 3 hours Problem 1. Prove or give a counter-example to the following statement: If M/L and L/K are algebraic extensions of fields, then M/K is algebraic.

More information

FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS

FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS Sairaiji, F. Osaka J. Math. 39 (00), 3 43 FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS FUMIO SAIRAIJI (Received March 4, 000) 1. Introduction Let be an elliptic curve over Q. We denote by ˆ

More information

Notes on p-divisible Groups

Notes on p-divisible Groups Notes on p-divisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete

More information

Higher Ramification Groups

Higher Ramification Groups COLORADO STATE UNIVERSITY MATHEMATICS Higher Ramification Groups Dean Bisogno May 24, 2016 1 ABSTRACT Studying higher ramification groups immediately depends on some key ideas from valuation theory. With

More information

RUDIMENTARY GALOIS THEORY

RUDIMENTARY GALOIS THEORY RUDIMENTARY GALOIS THEORY JACK LIANG Abstract. This paper introduces basic Galois Theory, primarily over fields with characteristic 0, beginning with polynomials and fields and ultimately relating the

More information

3 Extensions of local fields

3 Extensions of local fields 3 Extensions of local fields ocal field = field complete wrt an AV. (Sometimes people are more restrictive e.g. some people require the field to be locally compact.) We re going to study extensions of

More information

ORAL QUALIFYING EXAM QUESTIONS. 1. Algebra

ORAL QUALIFYING EXAM QUESTIONS. 1. Algebra ORAL QUALIFYING EXAM QUESTIONS JOHN VOIGHT Below are some questions that I have asked on oral qualifying exams (starting in fall 2015). 1.1. Core questions. 1. Algebra (1) Let R be a noetherian (commutative)

More information

Notes on graduate algebra. Robert Harron

Notes on graduate algebra. Robert Harron Notes on graduate algebra Robert Harron Department of Mathematics, Keller Hall, University of Hawai i at Mānoa, Honolulu, HI 96822, USA E-mail address: rharron@math.hawaii.edu Abstract. Graduate algebra

More information

Lemma 1.1. The field K embeds as a subfield of Q(ζ D ).

Lemma 1.1. The field K embeds as a subfield of Q(ζ D ). Math 248A. Quadratic characters associated to quadratic fields The aim of this handout is to describe the quadratic Dirichlet character naturally associated to a quadratic field, and to express it in terms

More information

Galois groups with restricted ramification

Galois groups with restricted ramification Galois groups with restricted ramification Romyar Sharifi Harvard University 1 Unique factorization: Let K be a number field, a finite extension of the rational numbers Q. The ring of integers O K of K

More information

7 Orders in Dedekind domains, primes in Galois extensions

7 Orders in Dedekind domains, primes in Galois extensions 18.785 Number theory I Lecture #7 Fall 2015 10/01/2015 7 Orders in Dedekind domains, primes in Galois extensions 7.1 Orders in Dedekind domains Let S/R be an extension of rings. The conductor c of R (in

More information

Extensions of Discrete Valuation Fields

Extensions of Discrete Valuation Fields CHAPTER 2 Extensions of Discrete Valuation Fields This chapter studies discrete valuation fields in relation to each other. The first section introduces the class of Henselian fields which are quite similar

More information

5. Kato s higher local class field theory

5. Kato s higher local class field theory ISSN 1464-8997 (on line) 1464-8989 (printed) 53 Geometry & Topology Monographs Volume 3: Invitation to higher local fields Part I, section 5, pages 53 60 5. Kato s higher local class field theory Masato

More information

Notes on Galois Theory

Notes on Galois Theory Notes on Galois Theory Math 431 04/28/2009 Radford We outline the foundations of Galois theory. Most proofs are well beyond the scope of the our course and are therefore omitted. The symbols and in the

More information

ON TAMELY RAMIFIED IWASAWA MODULES FOR THE CYCLOTOMIC p -EXTENSION OF ABELIAN FIELDS

ON TAMELY RAMIFIED IWASAWA MODULES FOR THE CYCLOTOMIC p -EXTENSION OF ABELIAN FIELDS Itoh, T. Osaka J. Math. 51 (2014), 513 536 ON TAMELY RAMIFIED IWASAWA MODULES FOR THE CYCLOTOMIC p -EXTENSION OF ABELIAN FIELDS TSUYOSHI ITOH (Received May 18, 2012, revised September 19, 2012) Abstract

More information

CLASS FIELD THEORY AND COMPLEX MULTIPLICATION FOR ELLIPTIC CURVES

CLASS FIELD THEORY AND COMPLEX MULTIPLICATION FOR ELLIPTIC CURVES CLASS FIELD THEORY AND COMPLEX MULTIPLICATION FOR ELLIPTIC CURVES FRANK GOUNELAS 1. Class Field Theory We ll begin by motivating some of the constructions of the CM (complex multiplication) theory for

More information

10 l-adic representations

10 l-adic representations 0 l-adic representations We fix a prime l. Artin representations are not enough; l-adic representations with infinite images naturally appear in geometry. Definition 0.. Let K be any field. An l-adic Galois

More information

9 Artin representations

9 Artin representations 9 Artin representations Let K be a global field. We have enough for G ab K. Now we fix a separable closure Ksep and G K := Gal(K sep /K), which can have many nonabelian simple quotients. An Artin representation

More information

KUMMER S CRITERION ON CLASS NUMBERS OF CYCLOTOMIC FIELDS

KUMMER S CRITERION ON CLASS NUMBERS OF CYCLOTOMIC FIELDS KUMMER S CRITERION ON CLASS NUMBERS OF CYCLOTOMIC FIELDS SEAN KELLY Abstract. Kummer s criterion is that p divides the class number of Q(µ p) if and only if it divides the numerator of some Bernoulli number

More information

Cover Page. The handle holds various files of this Leiden University dissertation.

Cover Page. The handle   holds various files of this Leiden University dissertation. Cover Page The handle http://hdl.handle.net/1887/20310 holds various files of this Leiden University dissertation. Author: Jansen, Bas Title: Mersenne primes and class field theory Date: 2012-12-18 Chapter

More information

Maximal Class Numbers of CM Number Fields

Maximal Class Numbers of CM Number Fields Maximal Class Numbers of CM Number Fields R. C. Daileda R. Krishnamoorthy A. Malyshev Abstract Fix a totally real number field F of degree at least 2. Under the assumptions of the generalized Riemann hypothesis

More information

Selected exercises from Abstract Algebra by Dummit and Foote (3rd edition).

Selected exercises from Abstract Algebra by Dummit and Foote (3rd edition). Selected exercises from Abstract Algebra by Dummit and Foote (3rd edition). Bryan Félix Abril 12, 2017 Section 14.2 Exercise 3. Determine the Galois group of (x 2 2)(x 2 3)(x 2 5). Determine all the subfields

More information

NOTES FOR DRAGOS: MATH 210 CLASS 12, THURS. FEB. 22

NOTES FOR DRAGOS: MATH 210 CLASS 12, THURS. FEB. 22 NOTES FOR DRAGOS: MATH 210 CLASS 12, THURS. FEB. 22 RAVI VAKIL Hi Dragos The class is in 381-T, 1:15 2:30. This is the very end of Galois theory; you ll also start commutative ring theory. Tell them: midterm

More information

Local Fields. Chapter Absolute Values and Discrete Valuations Definitions and Comments

Local Fields. Chapter Absolute Values and Discrete Valuations Definitions and Comments Chapter 9 Local Fields The definition of global field varies in the literature, but all definitions include our primary source of examples, number fields. The other fields that are of interest in algebraic

More information

Primes of the form X² + ny² in function fields

Primes of the form X² + ny² in function fields Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 2010 Primes of the form X² + ny² in function fields Piotr Maciak Louisiana State University and Agricultural and

More information

CLASS FIELD THEORY FOR NUMBER FIELDS AND COMPLEX MULTIPLICATION

CLASS FIELD THEORY FOR NUMBER FIELDS AND COMPLEX MULTIPLICATION CLASS FIELD THEORY FOR NUMBER FIELDS AND COMPLEX MULTIPLICATION GWYNETH MORELAND Abstract. We state the main results of class field theory for a general number field, and then specialize to the case where

More information

Quasi-reducible Polynomials

Quasi-reducible Polynomials Quasi-reducible Polynomials Jacques Willekens 06-Dec-2008 Abstract In this article, we investigate polynomials that are irreducible over Q, but are reducible modulo any prime number. 1 Introduction Let

More information

FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS.

FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. Let A be a ring, for simplicity assumed commutative. A filtering, or filtration, of an A module M means a descending sequence of submodules M = M 0

More information

Explicit Formulas for Hilbert Pairings on Formal Groups

Explicit Formulas for Hilbert Pairings on Formal Groups CHAPTER 8 Explicit Formulas for Hilbert Pairings on Formal Groups The method of the previous chapter possesses a valuable property: it can be relatively easily applied to derive explicit formulas for various

More information

Primes of the Form x 2 + ny 2

Primes of the Form x 2 + ny 2 Primes of the Form x 2 + ny 2 Steven Charlton 28 November 2012 Outline 1 Motivating Examples 2 Quadratic Forms 3 Class Field Theory 4 Hilbert Class Field 5 Narrow Class Field 6 Cubic Forms 7 Modular Forms

More information

Galois Theory. This material is review from Linear Algebra but we include it for completeness.

Galois Theory. This material is review from Linear Algebra but we include it for completeness. Galois Theory Galois Theory has its origins in the study of polynomial equations and their solutions. What is has revealed is a deep connection between the theory of fields and that of groups. We first

More information

ALGEBRA 11: Galois theory

ALGEBRA 11: Galois theory Galois extensions Exercise 11.1 (!). Consider a polynomial P (t) K[t] of degree n with coefficients in a field K that has n distinct roots in K. Prove that the ring K[t]/P of residues modulo P is isomorphic

More information

Lecture 8: The Field B dr

Lecture 8: The Field B dr Lecture 8: The Field B dr October 29, 2018 Throughout this lecture, we fix a perfectoid field C of characteristic p, with valuation ring O C. Fix an element π C with 0 < π C < 1, and let B denote the completion

More information

Imaginary Quadratic Fields With Isomorphic Abelian Galois Groups

Imaginary Quadratic Fields With Isomorphic Abelian Galois Groups Imaginary Quadratic Fields With Isomorphic Abelian Galois Groups Universiteit Leiden, Université Bordeaux 1 July 12, 2012 - UCSD - X - a Question Let K be a number field and G K = Gal(K/K) the absolute

More information

The Galois group of a polynomial f(x) K[x] is the Galois group of E over K where E is a splitting field for f(x) over K.

The Galois group of a polynomial f(x) K[x] is the Galois group of E over K where E is a splitting field for f(x) over K. The third exam will be on Monday, April 9, 013. The syllabus for Exam III is sections 1 3 of Chapter 10. Some of the main examples and facts from this material are listed below. If F is an extension field

More information

CYCLOTOMIC FIELDS CARL ERICKSON

CYCLOTOMIC FIELDS CARL ERICKSON CYCLOTOMIC FIELDS CARL ERICKSON Cyclotomic fields are an interesting laboratory for algebraic number theory because they are connected to fundamental problems - Fermat s Last Theorem for example - and

More information

Notes on Field Extensions

Notes on Field Extensions Notes on Field Extensions Ryan C. Reich 16 June 2006 1 Definitions Throughout, F K is a finite field extension. We fix once and for all an algebraic closure M for both and an embedding of F in M. When

More information

Chapter 8. P-adic numbers. 8.1 Absolute values

Chapter 8. P-adic numbers. 8.1 Absolute values Chapter 8 P-adic numbers Literature: N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd edition, Graduate Texts in Mathematics 58, Springer Verlag 1984, corrected 2nd printing 1996, Chap.

More information

SEPARABLE EXTENSIONS OF DEGREE p IN CHARACTERISTIC p; FAILURE OF HERMITE S THEOREM IN CHARACTERISTIC p

SEPARABLE EXTENSIONS OF DEGREE p IN CHARACTERISTIC p; FAILURE OF HERMITE S THEOREM IN CHARACTERISTIC p SEPARABLE EXTENSIONS OF DEGREE p IN CHARACTERISTIC p; FAILURE OF HERMITE S THEOREM IN CHARACTERISTIC p JIM STANKEWICZ 1. Separable Field Extensions of degree p Exercise: Let K be a field of characteristic

More information

School of Mathematics and Statistics. MT5836 Galois Theory. Handout 0: Course Information

School of Mathematics and Statistics. MT5836 Galois Theory. Handout 0: Course Information MRQ 2017 School of Mathematics and Statistics MT5836 Galois Theory Handout 0: Course Information Lecturer: Martyn Quick, Room 326. Prerequisite: MT3505 (or MT4517) Rings & Fields Lectures: Tutorials: Mon

More information

Lectures on Class Field Theory

Lectures on Class Field Theory Helmut Hasse Lectures on Class Field Theory April 7, 2004 v Translation, annotations and additions by Franz Lemmermeyer and Peter Roquette vi Preface Preface to the First Edition I have given the lectures

More information

1 The Galois Group of a Quadratic

1 The Galois Group of a Quadratic Algebra Prelim Notes The Galois Group of a Polynomial Jason B. Hill University of Colorado at Boulder Throughout this set of notes, K will be the desired base field (usually Q or a finite field) and F

More information

Identify F inside K and K. Then, by the claim, we have an embedding σ : K K such that σ F = id F. Since σ(k)/f and K/F are isomorphic extensions, the

Identify F inside K and K. Then, by the claim, we have an embedding σ : K K such that σ F = id F. Since σ(k)/f and K/F are isomorphic extensions, the 11. splitting field, Algebraic closure 11.1. Definition. Let f(x) F [x]. Say that f(x) splits in F [x] if it can be decomposed into linear factors in F [x]. An extension K/F is called a splitting field

More information

ALGEBRA EXERCISES, PhD EXAMINATION LEVEL

ALGEBRA EXERCISES, PhD EXAMINATION LEVEL ALGEBRA EXERCISES, PhD EXAMINATION LEVEL 1. Suppose that G is a finite group. (a) Prove that if G is nilpotent, and H is any proper subgroup, then H is a proper subgroup of its normalizer. (b) Use (a)

More information

Infinite Galois theory

Infinite Galois theory Treball final de grau GRAU DE MATEMÀTIQUES Facultat de Matemàtiques i Informàtica Universitat de Barcelona Infinite Galois theory Autor: Ignasi Sánchez Rodríguez Director: Realitzat a: Dra. Teresa Crespo

More information

Algebra Exam Fall Alexander J. Wertheim Last Updated: October 26, Groups Problem Problem Problem 3...

Algebra Exam Fall Alexander J. Wertheim Last Updated: October 26, Groups Problem Problem Problem 3... Algebra Exam Fall 2006 Alexander J. Wertheim Last Updated: October 26, 2017 Contents 1 Groups 2 1.1 Problem 1..................................... 2 1.2 Problem 2..................................... 2

More information

but no smaller power is equal to one. polynomial is defined to be

but no smaller power is equal to one. polynomial is defined to be 13. Radical and Cyclic Extensions The main purpose of this section is to look at the Galois groups of x n a. The first case to consider is a = 1. Definition 13.1. Let K be a field. An element ω K is said

More information

NUNO FREITAS AND ALAIN KRAUS

NUNO FREITAS AND ALAIN KRAUS ON THE DEGREE OF THE p-torsion FIELD OF ELLIPTIC CURVES OVER Q l FOR l p NUNO FREITAS AND ALAIN KRAUS Abstract. Let l and p be distinct prime numbers with p 3. Let E/Q l be an elliptic curve with p-torsion

More information

Cover Page. The handle holds various files of this Leiden University dissertation

Cover Page. The handle   holds various files of this Leiden University dissertation Cover Page The handle http://hdl.handle.net/1887/25833 holds various files of this Leiden University dissertation Author: Palenstijn, Willem Jan Title: Radicals in Arithmetic Issue Date: 2014-05-22 Chapter

More information

Algebra Ph.D. Entrance Exam Fall 2009 September 3, 2009

Algebra Ph.D. Entrance Exam Fall 2009 September 3, 2009 Algebra Ph.D. Entrance Exam Fall 2009 September 3, 2009 Directions: Solve 10 of the following problems. Mark which of the problems are to be graded. Without clear indication which problems are to be graded

More information

On the image of noncommutative local reciprocity map

On the image of noncommutative local reciprocity map On the image of noncommutative local reciprocity map Ivan Fesenko 0. Introduction First steps in the direction of an arithmetic noncommutative local class field theory were described in [] as an attempt

More information

Algebraic proofs of Dirichlet s theorem on arithmetic progressions

Algebraic proofs of Dirichlet s theorem on arithmetic progressions Charles University in Prague Faculty of Mathematics and Physics BACHELOR THESIS Martin Čech Algebraic proofs of Dirichlet s theorem on arithmetic progressions Department of Algebra Supervisor of the bachelor

More information

Absolute Values and Completions

Absolute Values and Completions Absolute Values and Completions B.Sury This article is in the nature of a survey of the theory of complete fields. It is not exhaustive but serves the purpose of familiarising the readers with the basic

More information

(1) A frac = b : a, b A, b 0. We can define addition and multiplication of fractions as we normally would. a b + c d

(1) A frac = b : a, b A, b 0. We can define addition and multiplication of fractions as we normally would. a b + c d The Algebraic Method 0.1. Integral Domains. Emmy Noether and others quickly realized that the classical algebraic number theory of Dedekind could be abstracted completely. In particular, rings of integers

More information

The Local Langlands Conjectures for n = 1, 2

The Local Langlands Conjectures for n = 1, 2 The Local Langlands Conjectures for n = 1, 2 Chris Nicholls December 12, 2014 1 Introduction These notes are based heavily on Kevin Buzzard s excellent notes on the Langlands Correspondence. The aim is

More information

Introduction to Arithmetic Geometry Fall 2013 Lecture #24 12/03/2013

Introduction to Arithmetic Geometry Fall 2013 Lecture #24 12/03/2013 18.78 Introduction to Arithmetic Geometry Fall 013 Lecture #4 1/03/013 4.1 Isogenies of elliptic curves Definition 4.1. Let E 1 /k and E /k be elliptic curves with distinguished rational points O 1 and

More information

Ohio State University Department of Mathematics Algebra Qualifier Exam Solutions. Timothy All Michael Belfanti

Ohio State University Department of Mathematics Algebra Qualifier Exam Solutions. Timothy All Michael Belfanti Ohio State University Department of Mathematics Algebra Qualifier Exam Solutions Timothy All Michael Belfanti July 22, 2013 Contents Spring 2012 1 1. Let G be a finite group and H a non-normal subgroup

More information

Local root numbers of elliptic curves over dyadic fields

Local root numbers of elliptic curves over dyadic fields Local root numbers of elliptic curves over dyadic fields Naoki Imai Abstract We consider an elliptic curve over a dyadic field with additive, potentially good reduction. We study the finite Galois extension

More information

On metacyclic extensions

On metacyclic extensions On metacyclic extensions Masanari Kida 1 Introduction A group G is called metacyclic if it contains a normal cyclic subgroup N such that the quotient group G/N is also cyclic. The category of metacyclic

More information

CSIR - Algebra Problems

CSIR - Algebra Problems CSIR - Algebra Problems N. Annamalai DST - INSPIRE Fellow (SRF) Department of Mathematics Bharathidasan University Tiruchirappalli -620024 E-mail: algebra.annamalai@gmail.com Website: https://annamalaimaths.wordpress.com

More information

18 The analytic class number formula

18 The analytic class number formula 18.785 Number theory I Lecture #18 Fall 2015 11/12/2015 18 The analytic class number formula The following theorem is usually attributed to Dirichlet, although he originally proved it only for quadratic

More information

Algebraic number theory, Michaelmas 2015

Algebraic number theory, Michaelmas 2015 Algebraic number theory, Michaelmas 2015 January 22, 2016 Contents 1 Discrete valuation rings 2 2 Extensions of Dedekind domains 7 3 Complete discrete valuation rings 8 4 The p-adic numbers 10 5 Extensions

More information

Algebraic Hecke Characters

Algebraic Hecke Characters Algebraic Hecke Characters Motivation This motivation was inspired by the excellent article [Serre-Tate, 7]. Our goal is to prove the main theorem of complex multiplication. The Galois theoretic formulation

More information

Part II Galois Theory

Part II Galois Theory Part II Galois Theory Definitions Based on lectures by C. Birkar Notes taken by Dexter Chua Michaelmas 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly)

More information

Introduction to Arithmetic Geometry Fall 2013 Lecture #7 09/26/2013

Introduction to Arithmetic Geometry Fall 2013 Lecture #7 09/26/2013 18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #7 09/26/2013 In Lecture 6 we proved (most of) Ostrowski s theorem for number fields, and we saw the product formula for absolute values on

More information

GEOMETRIC CLASS FIELD THEORY I

GEOMETRIC CLASS FIELD THEORY I GEOMETRIC CLASS FIELD THEORY I TONY FENG 1. Classical class field theory 1.1. The Artin map. Let s start off by reviewing the classical origins of class field theory. The motivating problem is basically

More information

8 Complete fields and valuation rings

8 Complete fields and valuation rings 18.785 Number theory I Fall 2017 Lecture #8 10/02/2017 8 Complete fields and valuation rings In order to make further progress in our investigation of finite extensions L/K of the fraction field K of a

More information

MAIN THEOREM OF GALOIS THEORY

MAIN THEOREM OF GALOIS THEORY MAIN THEOREM OF GALOIS THEORY Theorem 1. [Main Theorem] Let L/K be a finite Galois extension. and (1) The group G = Gal(L/K) is a group of order [L : K]. (2) The maps defined by and f : {subgroups of G}!

More information