# Galois Theory of Cyclotomic Extensions

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1 Galois Theory of Cyclotomic Extensions Winter School 2014, IISER Bhopal Romie Banerjee, Prahlad Vaidyanathan

2 I. Introduction 1. Course Description The goal of the course is to provide an introduction to algebraic number theory, which is essentially concerned with understanding algebraic field extensions of the field of rational numbers, Q. We will begin by reviewing Galois theory: 1.1. Rings and Ideals, Field Extensions 1.2. Galois Groups, Galois Correspondence 1.3. Cyclotomic extensions We then discuss Ramification theory: 1.1. Dedekind Domains 1.2. Inertia groups 1.3. Ramification in Cyclotomic Extensions 1.4. Valuations This will finally lead to a proof of the Kronecker-Weber Theorem, which states that If Q L is a finite Galois extension whose Galois group is abelian, then n N such that L Q(ζ n ), where ζ n denotes a primitive n th root of unity 2. Pre-requisites A first course in Galois theory. Some useful books are : 2.1. Ian Stewart, Galois Theory (3rd Ed.), Chapman & Hall (2004) 2.2. D.J.H. Garling, A Course in Galois Theory, Camridge University Press (1986) 2.3. D.S. Dummit, R.M. Foote, Abstract Algebra (2nd Ed.), John Wiley and Sons (2002) 2

3 3. Reference Material 3.1. M.J. Greenberg, An Elementary Proof of the Kronecker-Weber Theorem, The American Mathematical Monthly, Vol. 81, No. 6 (Jun.-Jul. 1974), pp S. Lang, Algebraic Number Theory, Addison-Wesley, Reading, Mass. (1970) 3.3. J. Neukrich, Algebraic Number Theory, Springer (1999) 4. Pre-requisites 4.1. Definition: (i) Rings (ii) Commutative Ring (iii) Units in a ring (iv) Field 4.2. Examples: (i) Q, R, C (ii) F p := Z/pZ for p Z prime (iii) Definition: Let k be a field (a) A polynomial over k (b) Polynomial ring k[x] (c) Degree of a polynomial 4.3. (Euclidean Division): Let k be a field. If f, g k[x] with g 0, t, r k[x] such that f = tg + r and deg(r) < deg(g) 4.4. k[x] is a principal ideal domain (PID) Definition: Let k be a field. (i) f g in k[x] (ii) GCD of f and g Theorem: Let k be a field. If f, g k[x], then the GCD d of f and g exists in k[x]. Furthermore, s, t k[x] such that d = sf + tg. We write d = (f, g) 4.7. Definition: (i) Ideal in a ring (ii) Maximal ideal (iii) Irreducible polynomial f k[x] 4.8. Theorem: For f k[x], the following are equivalent: (i) f is irreducible in k[x] 3

4 (ii) The ideal (f) is a maximal ideal (iii) k[x]/(f) is a field 4.9. Examples: (i) Polynomials of degree 1 are automatically irreducible (ii) x 2 2 is irreducible in Q[x], but not R[x] (iii) Polynomial of degree 2 or 3 is irreducible in k[x] iff it does not contain a root in k (Exercise) (Gauss Lemma): Let f Z[x] be monic, then f is irreducible in Z[x] iff it is irreducible in Q[x] (Rational Root theorem): Let f(x) = a 0 + a 1 x a n x n Z[x] have a root p/q Q where (p, q) = 1. Then (i) p a 0 and q a n (ii) In particular, if f is monic, then every rational root of f must be an integer (Eisenstein s Criterion): Let f(x) = a 0 + a 1 x a n x n Z[x]. Suppose p Z prime such that (i) p a i for all i {0, 1,..., n 1} (ii) p a n (iii) p 2 a 0 Then f is irreducible in Q[x] (Reduction (mod p)): Let f(x) = a 0 + a 1 x a n x n Z[x] and p Z prime such that p a n. If f is irreducible in Z p [x], then f is irreducible in Q[x]. (The converse is not true: f(x) = x Z[x] is reducible in Z 2 [x]) (Fundamental Theorem of Algebra): For any non-constant f C[x], α C such that f(α) = 0 4

5 II. Field Extensions 1. Simple Extensions 1.1. Remark: In this course, we will consider two kinds of fields : (i) Finite fields, such as F p = Z/pZ, p Z prime (ii) Subfields of C (which are necessarily infinite), such as Q, R, C We will begin by only considering subfields of C, and discuss finite fields later Definition: (i) Field extension k L (ii) Smallest field k(x) generated by a field k C and a set X C. (iii) A field extension k L is called simple if α L such that L = k(α). α is called a primitive element of the field extension. Note: The primitive element may not be unique (See Example 1.3(ii)) 1.3. Examples: (i) Q R, Q C are field extensions, but neither are simple. (Proof later) (ii) R C is a simple extension. C = R(i). Note that C = R(i + 1) as well, so the primitive element may not be unique. (iii) Every subfield k C contains Q. So Q k is a field extension. (Exercise) (iv) Let F = {a + b 2 : a, b Q}, then (a) F is a field (b) Hence, Q F is a field extension (c) Note that F = Q( 2) (v) Let K = Q( 2, 3), then K = Q( 2 + 3) and is hence a simple extension (vi) Let k be a field and f k[x] be irreducible. Set L = k[x]/(f), then k L is a field extension Definition/Remark: Let k C be a field and α C (i) α is said to be algebraic over k if 0 f k[x] such that f(α) = 0. (ii) α is said to be transcendental over k if it is not algebraic Examples: (i) If α k, then α is algebraic over k 5

6 (ii) 2 is algebraic over Q (iii) π is transcendental over Q (without proof) (iv) π is algebraic over R 1.6. Theorem: Let k L be a field extension and α L be algebraic over k. Then unique polynomial f k[x] such that (i) f is monic and irreducible (ii) f(α) = 0 Furthermore, if g k[x] is any polynomial, then g(α) = 0 iff f g in k[x]. This is called the minimal polynomial of α over k and is denoted by m α := m α,k. Proof. Let I = {g k[x] : g(α) = 0} Then by hypothesis, I {0}. Also, I is clearly an ideal in k[x]. Since k[x] is a PID, f I such that I = (f) By dividing by the leading coefficient, we may assume f is monic. (i) f is irreducible: If f = gh, then 0 = f(α) = g(α)h(α). Since C is a field, either g(α) = 0 or h(α) = 0. Assume WLOG that g(α) = 0, then g I to f g. However, g f as well, so g = cf for some constant c k. Hence, deg(h) = 0, which means that h k (ii) Suppose g k[x] is irreducible, monic and satisfies g(α) = 0. Then, g I and so f g. But g is irreducible, so f = cg for some c k. They are both monic, so c = 1. Hence we get uniqueness as well. (iii) Since f is irreducible, I is a maximal ideal, and so k[x]/i is a field. Define ϕ : k[x] C by f f(α) Then ϕ is a homomorphism and I = ker(ϕ). Thus, Image(ϕ) is a field. (iv) Clearly, k Image(ϕ) and α Image(ϕ). So k(α) Image(ϕ) Conversely, if β Image(ϕ), then f k[x] such that β = f(α). β k(α). Hence, Image(ϕ) = k(α) Thus, 1.7. Examples: (i) If α k, then m α (x) = x α (ii) If k = Q, α = 2, then m α (x) = x 2 2 6

7 (iii) If k = R, α = 2, then m α (x) = x 2 (iv) If k = Q, ζ p = e 2πi/p, where p is a prime, then m ζp (x) = Φ p (x) = x p 1 + x p x Definition: Let k L 1 and k L 2 be field extensions (i) A homomorphism of field extensions is a field homomorphism ϕ : L 1 L 2 such that ϕ(α) = α for all α k. (ii) An isomorphism of field extensions is a bijective homomorphism. If such an isomorphism exists, we write L 1 =k L Theorem: Let k C be a field and α C be algebraic over k. Then k[x]/(m α ) = k k(α) Proof. Example 1.3.(vi) and Theorem Corollary: Let k L be a field extension and α, β L be algebraic over k with the same minimal polynomial. Then there is an isomorphism of field extensions k(α) = k k(β) which sends α β. Exercises 1.1. Let k be a field and let f k[x] be of degree 2 or 3. Prove that f is irreducible in k[x] iff f does not have a root in k (i) Prove that f(x) is irreducible in Z[x] iff f(x + 1) is irreducible. (ii) Let p Z be a prime, then prove that, for any 1 k p 1, ( ) p p k (iii) Use (i) and (ii) to prove that is irreducible in Z[x] Φ p (x) := xp p 1 1 x 1 = 1.3. Let f(x) = x Z[x]. Use Part (i) of the previous problem to prove that f is irreducible in Z[x] 1.4. Let k be a field. Define ϕ : Z k by n n 1. (i) Prove that ker(ϕ) is either trivial or p Z such that ker(ϕ) = pz. i=0 x i 7

8 (ii) If k C is any field, then prove that Q k (iii) If k is a finite field, then prove that p Z prime such that Z p k. Definition: We say a field has characteristic n Z iff ker(ϕ) = nz. We write char(k) = n Note: By the proof of (i), n must either be 0 or a prime. Hence, (i) If k C, then char(k) = 0 (ii) If k is a finite field, then char(k) = p for some prime p Z 1.5. (i) If f(x) Z[x] is irreducible, then prove that f( x) Z[x] is irreducible. (ii) Determine the minimal polynomial of α = e πi/5 over Q Let ϕ : Q C be a homomorphism of fields, then prove that ϕ(x) = x x Q Note: Since ϕ is a field homomorphism, ϕ(1) = 1 must hold by definition Determine all possible homomorphisms ϕ : Q( 2) C [Hint: Use the previous problem to prove that any such homomorphism is completely determined by ϕ( 2). Now determine the possible values of ϕ( 2)] 1.8. Let ω = e 2πi/3. Prove that there is an isomorphism 1.9. Is the set a field? Q( 3 2) = Q Q(ω 3 2) L := {a + bπ : a, b Q} (i) Let k L be a field extension, then L is a k-vector space. (ii) If k L 1 and k L 2 are two extensions, then a homomorphism ϕ : L 1 L 2 of k extensions is a k-linear map of vector spaces. Definition: Let k L be a field extension (i) The degree of the extension, denoted by [L : k], is the dimension of the k-vector space L. (ii) The field extension is called finite if [L : k] < Prove that (i) Q Q( 2) is a finite extension of degree 2. (ii) Q R is not a finite extension. [Hint: Q is countable, and R is not] If k is a finite field, then prove that p N prime and n N such that k = p n Let Q L be a field extension of degree 2 with Q-basis {1, α}. (i) Prove that α satisfies a polynomial f Q[x] of degree 2. (ii) Prove that r Q such that L = Q( r) [Hint: Use the quadratic formula] 8

9 2. Finite and Algebraic Extensions 2.1. Recall: If k L is a field extension, then (i) L is a k-vector space (ii) The dimension of L over k is called the degree of the extension, and is denoted by [L : k] (iii) k L is called a finite extension iff [L : k] < 2.2. Theorem: Let k C be a field and α C be algebraic over k. Let m α k[x] be the minimal polynomial of α over k, and let n = deg(m α ). Then (i) {1, α, α 2,..., α n 1 } is a basis for k(α) over k (ii) In particular, [k(α) : k] = deg(m α ) < Proof. Let S = {1, α, α 2,..., α n 1 }. (i) S is linearly independent: Suppose a 0, a 1,..., a n 1 k not all zero such that n 1 a i α i = 0 i=0 Then α satisfies the non-zero polynomial f(x) = a 0 + a 1 x a n 1 x n 1 k[x]. This contradicts the minimality of n = deg(m α ). (ii) S spans k(α): Suppose β k(α), then by Theorem 1.9, f k[x] such that β = f(α). By Euclidean division, t, r k[x] such that f = tm α + r, where deg(r) < n Clearly, f(α) = r(α), so β = r(α) and deg(r) < n. Hence, β Span(S) Examples: (i) Q( 2) = {a + b 2 : a, b Q} (This agrees with Example 1.3(iv)) (ii) Q( 3 2) = {a + b2 1/3 + c2 2/3 : a, b, c Q} (iii) Let p Z be a prime number and ζ p := e 2πi/p C, then [Q(ζ p ) : Q] = deg(φ p ) = p (Tower Law) If k F and F L are two field extensions, then [L : k] = [L : F ][F : k] 9

10 Proof. Suppose [F : k] is infinite, then [L : k] is infinite. Similarly, if [L : F ] is infinite, so is [L : k]. Hence, we may assume that the RHS is finite. Suppose is a k-basis for F, and S 1 := {x 1, x 2,..., x n } S 2 := {y 1, y 2,..., y m } is an F -basis for L. We want to show that has nm elements and is a k-basis for L. S := {x i y j : 1 i n, 1 j m} (i) S has nm elements : Suppose not, then x i1 y j1 = x i2 y j2. If y j1 = y j2, then x i1 = x i2 and we are done. If not, then x i1 y j1 x i2 y j2 = 0 But {y j1, y j2 } is F -linearly independent, so x i1 = x i2 = 0. This contradicts the k-linear independence of {x i1 }. (ii) S spans L over k: Suppose β L, then a 1, a 2,..., a m F such that m β = a j y j i=1 But a j F implies that α i,j k such that n a j = α i,j x i Hence, β = i,j α i,jx i y j (iii) S is k-linearly independent in L: Suppose α i,j k such that α i,j x i y j = 0 i,j i=1 Collect the coefficient of y j into a j and write m j=1 a jy j = 0 where a j F. However this would imply that a j = 0 for all j. Again, the k-linear independence of {x i } implies that α i,j = 0 for all i, j Example: Let f(x) = x 3 + 6x + 2 Q[x]. Then f is irreducible over Q( 4 2) Proof. Suppose not, then by Exercise 1.1, f must have a root α Q( 4 2). Hence, we have a tower Q Q(α) Q( 4 2) However, [Q(α) : Q] = deg(m α ). But f is irreducible over Q by Eisenstein, so it must be the minimal polynomial of α. Hence, [Q(α) : Q] = 3, which, by the Tower Law, would imply that 3 4. This is a contradiction. 10

11 2.6. Corollary: Let k F 1 and k F 2 be field extensions (all contained in C). Let L denote the smallest field containing both F 1 and F 2. Then (i) [L : F 2 ] [F 1 : k] (ii) [L : k] [F 1 : k][f 2 : k] L is called the compositum of F 1 and F 2 and is denoted by F 1 F 2 Proof. (i) Again, assume [F 1 : k] and [F 2 : k] are finite. Suppose S := {x 1, x 2,..., x n } is a k-basis for F 2 which contains 1. Claim: Let F 3 = Span F1 (S) L = Span F1 (S) = { n a i x i : a i F 1 } i=1 (a) F 3 L since F 1 L and S L and L is a field. (b) F 1 F 3 since 1 S (c) F 2 F 3 since F 2 = Span k (S) and k F 1 (d) F 3 is a field: A. F 3 is clearly closed under addition since it is a vector space. B. If x = n i=1 α ix i and y = n j=1 β jx j with α i, β j F 1, then xy = i,j α i β j x i x j Now, x i x j F 2 = Span k (S). Since k F 1 it follows that, xy Span F1 (S) C. If 0 x = n i=1 α ix i F 3. Then consider the map T : F 3 F 3 given by y xy Then T is clearly F 1 -linear, and is injective (why?). Since [F 3 : F 1 ] <, it follows that T is surjective. Hence, y F 3 such that xy = 1. (e) Since F 3 is a field and F 1 and F 2 are contained in F 3, it follows that L F 3. By part (a), we have that L = F 3 = Span F1 (S) Hence [L : F 2 ] S = [F 1 : k] (ii) Part (ii) follows from (i) and the fact that [L : k] = [L : F 2 ][F 2 : k] 11

12 2.7. Definition: A field extension k L is said to be algebraic if every α L is algebraic over k Theorem: (i) If k L is finite extension, then it is algebraic. (ii) If α L is algebraic over k, then k k(α) is algebraic. Proof. (i) Suppose k L is a finite extension of degree n. Let α L, then the set S = {1, α, α 2,..., α n } has n+1 elements and so it must be k-linearly dependent. Hence, a 0, a 1,..., a n k not all zero such that n a i α i = 0 and so α satisfies the non-zero polynomial f(x) = n i=0 a ix i k[x] (ii) Follows from part (i) and Theorem 2.1 i= Definition: An extension k L is said to be finitely generated if α 1, α 2,... α n L such that L = k(α 1, α 2,..., α n ) In other words, L is the smallest field containing k and the set {α 1, α 2,..., α n } Theorem: k L is a finite extension iff it is algebraic and finitely generated. Proof. (i) Suppose k L is a finite extension, then let S be a k-basis for L. Then L = k(s) and so k L is finitely generated. Also, by Theorem 3.2, k L is algebraic. (ii) Suppose k L is finitely generated and algebraic, then {α 1, α 2,..., α n } C such that L = k(α 1, α 2,..., α n ) By hypothesis, α i is algebraic over k, and so [k(α i ) : k] < By Theorem 2.6(ii) and induction, it follows that [L : k] n [k(α i ) : k] < i=1 Hence, k L is finite extension. 12

13 2.11. Example: Let k C be a field and set Then (i) F is a field (ii) k F is an algebraic extension (iii) If k = Q, then k F is infinite F = {α C : α is algebraic over k} Theorem: Let k C be any field and set F to be the field of numbers that are algebraic over k. For every non-constant f F [x], α F such that f(α) = 0. Proof. Suppose f(x) = a 0 + a 1 x a n x n F [x] is a non-constant polynomial, then α C such that f(α) = 0. But then, α is algebraic over F. We want to show that α is algebraic over k. So consider F 0 = k(a 0, a 1,..., a n ) Each a i F, so a i is algebraic over k. By Theorem 2.10 [F 0 : k] < Also, α is algebraic over F 0. So by Theorem 2.2, So by the Tower Law [F 0 (α) : F 0 ] < [F 0 (α) : k] < which implies that α is algebraic over k by Theorem 2.1. definition Remark: (i) We say that F is algebraically closed. Hence, α F by (ii) F is the smallest field that contains k and is algebraically closed (Why?) (iii) F is called the algebraic closure of k, and is denoted by k (iv) If k is any field (even a finite field), we can construct a field L such that (a) k L is algebraic (b) L is algebraically closed (c) If k M is any field extension satisfying the above two properties, then an injective homomorphism σ : L M. In other words, L is unique upto isomorphism. algebraic closure of k, and is denoted by k This field L is called the 13

14 Exercises 2.1. Prove that (i) 3 / Q( 2) (ii) [Q( 2, 3) : Q] = If k L is a finite extension such that [L : k] is prime, then (i) If F is a field such that k F L, then prove that either F = k or F = L (ii) Prove that α L such that L = k(α) (ie. k L is a simple extension) 2.3. Suppose k F 1 and k F 2 are two field extensions whose degrees are relatively prime. Then prove that [F 1 F 2 : k] = [F 1 : k][f 2 : k] 2.4. Let p Z be prime and let ζ p := e 2πi/p. Determine [Q(2 1/p, ζ p ) : Q] 2.5. Let k C be a field and α C be algebraic over k, and suppose ϕ : k(α) C is a field homomorphism such that ϕ k = id k. Let β := ϕ(α) Then prove that (i) For any f k[x], ϕ(f(α)) = f(β) (ii) β is algebraic over k (iii) β and α have the same minimal polynomial over k Use the previous problem and Corollary 1.10 to answer the following question: If α C is algebraic over Q, determine the number of homomorphisms 2.7. For each n N, define (i) Prove that A n is countable. (ii) Prove that B n is countable. (iii) Prove that Q is countable. ϕ : Q(α) C A n := {f Q[x] : deg(f) n} B n := f A n {α C : f(α) = 0} Conclude that there exist real numbers that are transcendental. 14

15 3. The Galois Group 3.1. Recall: Let k C be a field, α C algebraic over k (i) If k k(α) is a finite extension, and { n 1 } k(α) = a i α i : a i k i=0 where n = deg(m α ) (ii) If ϕ : k(α) C is a homomorphism such that ϕ k = id k, then n 1 a i α i ) = a i ϕ(α) i n 1 ϕ( i=0 So, ϕ is completely determined by the complex number ϕ(α). By Exercise 2.5, ϕ(α) must be a root of the minimial polynomial m α of α. Hence, if i=0 S = {k-homomorphisms ϕ : k(α) C}, and T = {roots of m α in C} Then we have a map µ : S T given by ϕ ϕ(α) We have shown above that this function µ is injective. (iii) By Corollary 1.10, if β T, then an isomorphism k(α) = k k(β). In particular, we get a k-homomorphism ϕ : k(α) C such that ϕ(α) = β. This proves that µ is surjective. Hence, {k-homomorphisms ϕ : k(α) C} = {roots of m α in C} 3.2. Examples: List all homomorphisms from k C: (i) k = Q: By Exercise 1.6, there is only one map, the inclusion (ii) k = Q( 2): By Exercise 1.7, there are two maps given by σ 1 : 2 2 σ 2 : 2 2 (iii) k = Q(ω): There are two maps given by σ 1 : ω ω σ 2 : ω ω 2 15

16 (iv) k = Q( 3 2): There are three maps given by σ 1 : σ 2 : σ 3 : ω ω (v) k = Q( 2, 3): There are atmost 4 maps given by σ 1 = id σ 2 = 2 2 and 3 3 σ 3 = 2 2 and 3 3 σ 4 = 2 2 and 3 3 (vi) k = Q( 3 2, ω): There are atmost 6 maps given by σ 1 = id L σ 2 = and ω ω 2 σ 3 = 3 2 ω 3 2 and ω ω σ 4 = 3 2 ω 3 2 and ω ω 2 σ 5 = 3 2 ω and ω ω σ 6 = 3 2 ω and ω ω Theorem: Let f k[x] be a monic irreducible polynomial, then all the roots of f in C are distinct (ie. f does not have any multiple roots. We say that such a polynomial is separable) Proof. Suppose β C is a multiple root of f, then (x β) 2 f in C[x]. Hence, f (β) = 0, whence f f in k[x] (since f must be the minimal polynomial of β over k). However, deg(f ) < deg(f), so this is impossible (Primitive Element Theorem): Let k L be a finite extension of subfields of C, then it is a simple extension. ie. θ L such that L = k(θ) 3.5. Corollary: Let k L be a finite extension of subfields of C, then the number of k-homomorphisms ϕ : L C = [L : k] 3.6. Definition: A field extension k L is said to be normal if every k-homomorphism ϕ : L C maps L into itself Lemma: If k L is a finite normal extension, then, for any k-homomorphism ϕ : L C, ϕ : L L is bijective. 16

17 Proof. Since k L is normal, ϕ(l) L. Hence we may consider ϕ : L L Note that ϕ is a k-linear transformation, and it is injective since it is a homomorphism of fields. Since [L : k] <, it must be surjective Definition: (i) A field extension that is both finite and normal is called a Galois extension. (ii) If k L is a Galois extension, then we define the Galois group of the extension as Gal k (L) = {k-homomorphisms ϕ : L C} 3.9. Examples: Note: Gal k (L) is a group under composition. (i) Gal k (k) = {id k } (ii) Gal Q (Q( 2)) = Z 2 (iii) Gal Q (Q(ω)) = Z 2 (iv) Gal Q (Q( 2, 3)) = Z 2 Z 2 Proof. Let L = Q( 2, 3). By Example 3.2, we have atmost 4 maps ϕ : L C. By Corollary 3.5, there are exactly 4 maps, so these must be all of them. For each of these maps, ϕ( 2) and ϕ( 3) belong to Q( 2, 3). Hence, by Lemma 3.7, they must be bijective as well. Hence, Gal k (L) = {σ 1, σ 2, σ 3, σ 4 } Since it is a group of order 4, Gal k (L) = Z 4 or Z 2 Z 2. However, σ 2 i = id L i Hence, Gal k (L) = Z 2 Z 2 (v) Gal Q (Q( 3 2, ω)) = S 3 Proof. Let L = Q( 3 2, ω). Again, there are exactly 6 homomorphisms ϕ : L C in Example 3.2. Also, for each of these homomorphisms ϕ( 3 2) and ϕ(ω) belong to L. Hence by Lemma 3.7, Gal k (L) = {σ i : 1 i 6} It is a group of order 6, so Gal k (L) = Z 6 or S 3. However, σ 3 σ 2 ( 3 2) σ 2 σ 3 ( 3 2) Hence, Gal k (L) is non-abelian, which forces it to be S 3 17

18 3.10. Lemma: Let k L be a finite normal extension. If F is an intermediate field k F L Then F L is a finite normal extension. Proof. (i) Clearly, [L : F ] [L : k] < (ii) If ϕ : L C is an F -homomorphism, then ϕ is also a k-homomorphism. Since k L is normal, ϕ(l) L Definition: Let k L be a field extension and G := Gal k (L) (i) If k F L is an intermediate field, then Gal F (L) < Gal k (L) (ii) If H < G, then L H := {x L : ϕ(x) = x ϕ H} L is called the fixed field of H Note: L H is a subfield of L containing k. (iii) We set F := {intermediate fields k F L} G := {subgroups H < G} Φ : F G Ψ : G F F Gal F (L) H L H Examples: (i) If k L is any field extension, and G = Gal k (L) (a) If H = {e} < G, then L H = L However, L G may not be equal to k (See below) (b) If F = L, then Gal F (L) = {e} If F = k, then Gal k (L) = G (ii) If k = Q, L = Q( 2, 3), then (a) If H = σ 3 where σ 3 : 2 2 and 3 3 Then L H = Q( 2) and Gal Q( 2) (L) = H 18

19 (b) If H = σ 4 where σ 4 : 2 2 and 3 3 Then L H = Q( 6) (iii) If k = Q, L = Q( 3 2, ω), then (a) If H = σ 2 where σ 2 : and ω ω 2 Then L H = Q( 3 2). Also, Gal Q( 3 2) (L) = H. Note that Q Q( 3 2) is not normal (Fundamental Theorem of Galois Theory - I): Let k L be a Galois extension of subfields of C with Galois group G. Then (i) Ψ and Φ are mutual inverses. So there is a one-to-one correspondence F G (ii) If F F is an intermediate field, then We visualize this by tower diagrams [F : k] = [Gal k (L) : Gal F (L)] L Gal k (L) = F Gal F (L) = k {e} (Fundamental Theorem of Galois Theory - II): Let k L be a Galois extension of subfields of C with Galois group G. Then (i) If F F, k F is normal iff Gal F (L) Gal k (L). (ii) In that case, the restriction map π : Gal k (L) Gal k (F ) is a well-defined, surjective, group homomorphism. (iii) ker(π) = Gal F (L) (iv) Hence, Gal k (L)/ Gal F (L) = Gal k (F ) 19

20 Exercises 3.1. If k L is a Galois extension with Galois group G. (i) If F is an intermediate field (k F L), then prove that Gal F (L) < G (ii) If H < G, then prove that L H is a field such that k L H L Let k L be a Galois extension with Galois group G (i) If F 1 and F 2 are intermediate fields such that F 1 F 2, then prove that Gal F2 (L) Gal F1 (L) (ii) If H 1 and H 2 are subgroups of G such that H 1 H 2, then prove that L H 2 L H Let L = Q( 3 2, ω), and G = Gal Q (L). Let F = Q( 3 2) and H = Gal F (L) (i) List all the elements of H from Example 3.2(vi) (ii) Prove that H is not normal in G Let L = Q( 3, 2) and let G = Gal Q (L) (i) Determine all the subgroups of G (ii) For each subgroup H from part (i), determine L H (Use Theorem 3.13(ii)) 3.5. Let L = Q( 3 2, ω) and G = Gal Q (L). (i) List all the subgroups of G from Example 3.2(vi) (ii) For each subgroup H from part (i), determine L H (Use Theorem 3.13(ii)) 3.6. Let L = Q( 4 2, i). (i) Prove that Q L is a normal extension of degree 8. (ii) Prove that Gal Q (L) has a subgroup H which is not normal. Note: There is only one group of order 8 which satisfies (ii). dihedral group of order 8, usually denoted by D 4 or D Let n N, ζ n = e 2πi/n and L = Q(ζ n ). (i) Prove that Q L is a finite extension of degree n 1. This is the (ii) If ϕ : L C is a homomorphism, then prove that 1 i n 1 such that ϕ(ζ n ) = ζ i n Conclude that Q L is a normal extension (iii) Use part (ii) to prove that Gal Q (L) is an abelian group Read Theorem 3.14 before attempting this problem: Let k L be a finite normal extension whose Galois group is abelian. (i) If F is an intermediate field, k F L, then prove that k F is a normal extension. (Compare this with Exercise 3.1) (ii) Prove that Gal k (F ) is an abelian group. 20

21 4. Finite Fields Note: All fields in this section will be finite of characteristic p > Remark: If L is a finite field of characteristic p > 0, then (i) F p L. Hence, L = p n for some n 1 (ii) We may construct an algebraically closed field, F p which will play the role that C played in the earlier discission. (iii) Note that the only homomorphism ϕ : F p F p is the inclusion map, so F p will play the role that Q played in the earlier discussion. (iv) As before, if k is a finite field and α F p, then there is a 1-1 correspondence {k-homomorphisms ϕ : k(α) F p } {roots of m α in F p } (Exercise 2.5 and Corollary 1.10 hold for finite fields without any change in the proof) 4.2. Theorem: If k is a finite field and f k[x] is irreducible, then it is separable. (ie. It has no multiple roots in F p ) 4.3. Corollary: If k is a finite field and α F p, then {k-homomorphisms ϕ : k(α) F p } = [k(α) : k] 4.4. Theorem: If L is a finite field, then L is cyclic (as a multiplicative group) Proof. By the Fundamental theorem of finite abelian groups, we can write L = Zd1 Z d2... Z dm such that d 1 d 2... d m. Let n = d m, then for any α L, we have α n = 1 Hence, α is a root of the polynomial x n 1 L[x] which has atmost n roots. Hence, d 1 d 2... d m = L n = d m Hence, m = 1 and L = Zn is cyclic Corollary: If k L is a finite field extension of finite fields, then k L is simple. Proof. The primitive element may be chosen to be any generator of L 4.6. Corollary: If k L is a finite extension of finite fields, then the number of k-homomorphisms ϕ : L F p = [L : k] 21

22 Proof. Write L = k(α) by Corollary 4.3. Now, as in 3.1, {k-homomorphisms ϕ : L F p } = { roots of m α in F p } But m α is separable, so the RHS is deg(m α ) = [L : k] 4.7. Theorem: If L is a finite field of characteristic p > 0 and cardinality p n, then L = {α F p : α pn = α} Proof. Let S be the set in the RHS, then S is the set of roots of the polynomial x pn x F p [x]. Hence, S p n Since L = p n, it suffices to prove that L S. Now if α L, then we may assume α 0. Now α L, which is a group of order p n 1. So α pn 1 = 1 α pn = α Hence, α S, which proves the L S and completes the proof Corollary: For any prime p N and any n N, there is a unique field of cardinality p n. We denote this field by F p n 4.9. Definition: As before, we make the following definitions : (i) A field extension k L of finite fields is called normal if, for every k- homomorphism ϕ : L F p, we have ϕ(l) L. (ii) If k L is such an extension, we write Gal k (L) = {k-homomorphisms ϕ : L F p } Note: As before, If k L is a finite normal extension, Gal k (L) is a group Theorem: If k L is a finite extension of finite fields, then it is normal. Proof. Suppose ϕ : L F p is a k-homomorphism, and β L, then we want to show that ϕ(β) L. By Theorem 4.7, Hence, ϕ(β) {α F p : α pn = α} = L β pn = β ϕ(β) pn = ϕ(β) Lemma: If L is a field of characteristic p > 0, then the map F : L L defined by x x p is a homomorphism, called the Frobenius map. Proof. (i) F is a map from L to F p, but F p L is a normal extension by 4.10, so F (L) = L. (ii) F is clearly multiplicative 22

23 (iii) We only need to check that if α, β L, then (α + β) p = α p + β p. However, this follows by the binomial theorem and the fact that ( ) p p 1 k p 1 k Theorem: Let L = F p n, then Gal Fp (L) is a cyclic group of order n generated by F Proof. We know that F : L L is a homomorphism, and every homomorphism fixes F p, so F Gal Fp (L). Also, So it suffices to prove that o(f ) = n. (i) For any α L, Gal Fp (L) = [L : F p ] = n F n (α) = α pn = α o(f ) n (ii) Suppose o(f ) = s n, then F s = id L. Then for any α L, Hence, α ps = α α is a root of x ps x L {roots of x ps x in F p } p s But L = p n, so n s, which proves the theorem Corollary: If k L is a finite extension of finite fields of characteristic p > 0, then Gal k (L) is a cyclic group genered by F j where j = [k : F p ] Proof. Let G = Gal k (L), then G < Gal Fp (L), which is a cyclic group generated by F. Hence, G = F j for some 1 j n. Now Hence, by the Tower Law o(f j ) = G = [L : k] j = Gal F p (L) G = [L : F p] [L : k] = [k : F p ] 23

24 Exercises 4.1. Let L = F 3 [x]/(x 2 + 1). (i) Prove that L is a field of cardinality 9. (ii) List down all the elements of L 4.2. Let f(x) = x 3 2 F 7 [x] and L = F 7 [x]/(f) (i) Prove that L is a field of cardinality 7 3 (ii) Determine which elements of L are the roots of f. [Hint: 2 3 = 1 in F 7 ] 4.3. Construct a field with 4 elements Let k be a finite field and f k[x] be an irreducible polynomial. (i) Prove that n N such that f (x pn x) in k[x] [Hint: Use Theorem 4.7 on the field k[x]/(f)] (ii) Conclude that f is separable. (This proves Theorem 4.2) 4.5. Fix n N and let (i) If d n, then prove that is a subfield of L L = {α F p : α pn = α} k = {α F p : α pd = α} (ii) Conversely, if k L is a subfield of L, then prove that d n such that k is given as in part (i). [Hint: Consider the tower F p k L] (iii) If k is as in part (i), determine Gal k (L) 4.6. Let L be a finite field. Prove that there is a 1-1 correspondence {subfields of L} {subgroups of Gal Fp (L)} Note: This proves Theorem 3.13 in the case of finite fields. Theorem 3.14 is not needed since all finite extensions of finite fields are normal Let k F 1 and k F 2 be Galois extensions of subfields of C. Prove that k F 1 F 2 is a Galois extension Let k F 1 and k F 2 be Galois extensions of subfields of C. Let L = F 1 F 2, and prove that (i) k L is a Galois extension (Use the primitive element theorem) (ii) Define a function by µ : Gal k (L) Gal k (F 1 ) Gal k (F 2 ) ϕ (ϕ F1, ϕ F2 ) Prove that µ is well-defined and injective. 24

25 5. Cyclotomic Extensions 5.1. Definition: Fix n N (i) ζ n = e 2πi/n (ii) µ n = {e 2πik/n : 0 k n 1} = ζ n. Note: µ n is a cyclic group of order n. (iii) Elements of µ n are called roots of unity. Generators of µ n are called primitive n th roots of unity. (iv) Q(ζ n ) = Q(µ n ) is called the n th cyclotomic field. Note: Q Q(µ n ) is a normal extension. (v) If G is a group, then Aut(G) = {ϕ : G G : ϕ is an isomorphism} Theorem: Let k C be any field, then (i) k k(ζ n ) is a finite normal extension. (ii) The map given by Γ : Gal k (k(ζ n )) Aut(µ n ) ϕ ϕ µn is a well-defined injective homomorphism. Proof. (a) If ϕ Gal k (k(ζ n )), we want to show that ϕ µn Aut(µ n ). A. Firstly, if ζ µ n, then ϕ(ζ) n = 1, so ϕ(ζ) µ n B. Now, ϕ µn is clearly a homomorphism from µ n to itself. C. Since ϕ is injective, ϕ µn is injective. Since µ n is finite, it is also surjective. (b) So Γ is a well-defined function. It is clearly a homomorphism since the operation on both groups is composition. (c) If Γ(ϕ) = id µn, then ϕ(ζ n ) = ζ n. Since ϕ Q = id Q, it would follow that ϕ = id Q(ζn) 5.3. Recall: (i) If R is a ring, R = {u R : v R such that uv = 1}. (ii) R is a group under multiplication, called the group of units of R. (iii) If R = Z n, then R = {a Z n : (a, n) = 1} Note: Z n is an abelian group. 25

26 5.4. Theorem: Aut(µ n ) = Z n Proof. If a Z such that (a, n) = 1, then define σ a : µ n µ n by ζ n ζn a (i) σ a is clearly a homomorphism of µ n (ii) Since (a, n) = 1, s, t Z such that sa + tn = 1. Hence, σ a (ζ s ) = ζ as = ζ as+tn = ζ Hence, σ a is surjective. (iii) Since µ n is finite, this means that σ a is injective as well. So σ a Aut(µ n ). (iv) If a b in Z n, then n (b a), so σ a (ζ) = ζ a = ζ b = σ b (ζ) Hence, σ a = σ b. So we get a well-defined map Θ : Z n Aut(µ n ) given by a σ a (v) Θ is a homomorphism since σ ab = σ a σ b (vi) Θ is injective: If σ a = id µn, then ζ a = ζ ζ a 1 = 1 n (a 1) a = 1 in Z n (vii) Θ is surjective: If ϕ : µ n µ n is an automorphism, then 1 j n such that ϕ(ζ) = ζ j. Since ϕ is surjective, t N such that ϕ(ζ t ) = ζ ζ tj = ζ 1 Hence, s Z such that tj + sn = 1. So (j, n) = 1 whence j Z n and ϕ = σ j = Θ(j) 5.5. Lemma: Let n N and ζ µ n be a primitive n th root of unity. ζ a is a primitive n th root of unity if and only if (a, n) = 1 Proof. Suppose (a, n) = 1, then s, t Z such that sa + tn = 1. Hence ζ n = ζ sa+tn n = (ζ a n) s ζ a n Hence, ζ a n = µ n. Conversely, if ζ j n is a primitive root of unity, then k N such that ζ jk n = ζ n. Hence, n (jk 1) and so (j, n) = 1. 26

27 5.6. Definition: n th Cyclotomic polynomial is defined as Φ n (x) = (x ζ) where the product is taken over all the primitive roots of unity. By Lemma 5.5, 5.7. Theorem: Φ n Q[x] Proof. Write deg(φ n ) = Z n Φ n (x) = a 0 + a 1 x a l x l If ψ Gal Q (Q(ζ n )) and a Z such that (a, n) = 1, then b Z such that (b, n) = 1 such that ψ(ζ a n) = ζ ab n Since (ab, n) = 1, we see that ψ permutes the roots of Φ n. Hence, ψ(φ n (x)) = (x ψ(ζ)) = (x ζ) = Φ n (x) Hence, ψ(a i ) = a i for all i. Hence, if G = Gal Q (Q(ζ n )), we have a i Q(ζ n ) G By the Fundamental Theorem of Galois Theory - I, this implies that a i Q Theorem: Φ n is irreducible in Q[x] 5.9. Corollary: Gal Q (Q(ζ n )) = Z n Proof. By 5.2 and 5.4, Gal Q (Q(ζ n )) = a subgroup of Z n However, Φ n is in Q[x], it satisfies ζ n, and it is both irreducible and monic. Hence, Φ n is the minimal polynomial of ζ n over Q, whence [Q(ζ n ) : Q] = deg(φ n ) = Z n Hence, the map Γ from Theorem 5.2 must be surjective as well Corollary: If p N is a prime, then Gal Q (Q(ζ p )) is a cyclic group of order p Corollary: If F is an intermediate field Q F Q(µ n ) such that Q F is Galois, then Gal Q (F ) is an abelian group. Proof. Since Q F is Galois, by Theorem 3.14, Gal F (Q(µ n )) Gal Q (Q(µ n )) and Gal Q (Q(µ n ))/ Gal F (Q(µ n )) = Gal Q (F ) Hence, Gal Q (F ) is a quotient of the abelian group Z n, and so it must be abelian (Kronecker-Weber Theorem) If Q F is any finite normal extension such that Gal Q (F ) is abelian, then n N such that F Q(µ n ). The proof of this statement will take up the rest of the course. 27

28 Exercises Read all the previous sections carefully and ask any questions you may have 5.1. If m, n N has lcm l, then (i) Prove that Q(ζ n ) Q(ζ l ) and Q(ζ m ) Q(ζ l ) (ii) Prove that s, t Z such that ζ l = ζmζ s n t (iii) Conclude that Q(ζ n )Q(ζ m ) = Q(ζ l ) 5.2. (i) For any n N, prove that x n 1 = Φ d (x) d n (ii) If p N is prime, then prove that Φ 2p (x) = Φ p ( x) (Compare with Exercise 1.5) p 1 Φ p 2(x) = k=0 x pk (iii) Determine Φ 8 (x) (Compare with Exercise 1.3) Recall the following facts: (i) Chinese Remainder theorem: If n = p e 1 1 p e p e k k of n with {p i } all distinct primes, then Z n = (Z e p 1 1 ) (Z e p 2 2 )... (Z e p k is the prime decomposition (ii) If G = G 1 G 2 is the direct product of two finite groups. Then G is cyclic iff both G 1 and G 2 are cyclic and ( G 1, G 2 ) = If p N is prime, then prove that Z p e = pe p e If n N is such that Z n is cyclic, then prove that n can be divisible by atmost one odd prime Let n 3 (i) List down the element of Z 8, and prove that it is not cyclic. (ii) Prove that there is a surjective map Z 2 n Z 2 n 1 (iii) Use induction and part (i) to prove that Z 2 n k ) is not cyclic 5.6. If n N such that Z n is cyclic, then prove that either n = 4 or n = 2 i p j for some odd prime p and i {0, 1} and j N {0} Note: The converse to the above statement is also true. 28

29 Bibliography [Stewart] Ian Stewart, Galois Theory (3rd Ed.) [Rotman] J. Rotman, Galois Theory (2nd Ed.) [Garling] DJH Garling, A Course in Galois Theory [E. Artin] Emil Artin and A.N. Milgram, Galois Theory [Greenberg] M.J. Greenberg, An Elementary Proof of the Kronecker-Weber Theorem, The American Mathematical Monthly, Vol. 81, No. 6 (Jun.-Jul. 1974), pp [Lang] S. Lang, Algebraic Number Theory, Addison-Wesley, Reading, Mass. (1970) [Neukrich] J. Neukrich, Algebraic Number Theory, Springer (1999) 29

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