# 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 )

Size: px
Start display at page:

Download "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 )"

Transcription

1 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 an abelian Galois extension. There exists an n such that K Q(ζ n ). Theorem 1.1 is equivalent to the following equality Q ab = Q(ζ n ) n=1 where Q ab denotes the maximal abelian extension (the field that contains all the abelian extensions of Q.) So basically theorem 1.1 says that the maximal abelian extension of Q is the compositum of the cyclotomic extensions of Q. Therefore it gives a classification of abelian extensions of Q. In general the abelian extensions of a number field can be classified by means of class field theory. In this paper we present a proof of theorem 1.1 without appealing to class field theory. A remarkable aspect of this work is that it makes use of the local-global principle. In other words we obtain theorem 1.1 from the following theorem: Theorem 1.2. Local Kronecker-Weber Theorem Let K/Q p be an abelian Galois extension. There exists an n such that K Q p (ζ n ) 2. Notations and Fundamental Theorems Throughout this paper p will denote a rational prime, Q p the completion of rational numbers with respect to p-adic valuation, K p the completion of a number field K with respect to one of its prime ideals p and ζ n a primitive nth root of unity. We start with basic facts and well known theorems from algebraic number theory. We give some of the proofs. Definition 2.1. Let K and L be finite extensions of Q (or Q p.) The smallest field containing K and L is called the compositum of K and L and denoted as KL. 1

2 2 NIZAMEDDIN H. ORDULU Theorem 2.2. Let K and L be finite Galois extensions of Q.Gal(KL/Q) is isomorphic to the subgroup {(φ, ψ) φ K L = ψ K L } of Gal(K/Q) Gal(L/Q). Similar argument holds for Q p. Proof Let G = Gal(KL/Q) and H = {(φ, ψ) φ K L = ψ K L }. Clearly the map Λ : G H, σ (σ K, σ L ) defines an injective homomorphism between G and H. We show that this homomorphism is indeed an isomorphism by showing that G = H. Let M = K L and let [M : Q] = m, [KL : K] = k and [KL : L] = l. Viewing A = Gal(KL/K) and B = Gal(KL/L) as subgroups of Gal(KL/M) one can easily show that A B = {id KL } and the fixed field of AB is M. It follows that [KL : M] = kl. So [K : M] = l and [L : M] = k. Combining with [M : Q] = m and simple counting shows that H = klm. But G = [KL : Q] = [KL : M][M : Q] = klm so we are done. Theorem 2.3. Let L/Q be an abelian Galois extension and let Gal(L/Q) m = G i. Then L = Similar argument holds for Q p. i=1 m L G i. i=1 Proof It suffices to prove for m = 2. Let L/Q = G 1 G 2. Then L G 1 L G 2 theorem 2.2 Gal(L G 1 L G 2 ) = G 1 G 2. From this the theorem follows. = Q. By Theorem 2.4. Let L/K be a finite Galois extension. (L and K can be number fields or local fields) Let p be a prime ideal of K. Then p factorizes in L as p = b e 1b e 2...b e g The number e is called the ramification index. The degree of the extension of the residue fields O L mod b 1 / O K mod p is denoted by f. If the degree of L/K is n then n = efg. (If K and L are local g = 1) p is said to be totally ramified in L if e = n and unramified if e = 1. (If K and L are local fields then we say L/K is unramified or totally ramified if e = 1 or e = n respectively. A number field extension is said to be unramified if all prime ideals are unramified.) Proof See any introductory Algebraic Number theory book or [S2] p. 101.

3 A SIMPLE PROOF OF KRONECKER-WEBER THEOREM 3 Definition 2.5. Let L/K be a Galois extension, p a prime of K, b a prime lying above p. The decomposition group D b of b is given by D b = {σ Gal(L/K) σ(b) = b}.(if L and K are local then D b is the whole Galois group.) The ramification group I b is defined as follows: I b = {σ D b σ(α) α (mod b) for all α O L } p is unramified in L I b and L I b is the largest such field among the intermediate fields of L/K. Theorem 2.6. Let L/K be a Galois extension of number fields. If p is a prime of K and L b /K p is the localization of L/K with respect to p, then Gal(L b /K p ) = D b and the inertia groups of b in both extensions are isomorphic. Proof There exist injections i 1 : K K p and i 2 : L L b. Certainly any element of Gal(L b /K p ) induces an automorphism of i 2 (L)/i 1 (K). Furthermore since i 1 (K) is dense in K p and i 2 (L) is dense in L b the restriction Σ : Gal(L b /K p ) Gal(i 2 (L)/i 1 (K)) is injective. Furthermore since any automorphism of L b /K p preserves b-adic absolute value the image of Σ must be in D b. Conversely if σ D b then one can extend σ uniquely to an automorphism of L b /K p. Theorem 2.7. Let K and L be finite Galois extensions of Q p and suppose that L/K is Galois. Then there is a surjective homomorphism between the inertia groups I L and I K of L and K. Proof Let M/Q p be the maximal unramified subextension of L/Q p. Then the maximal unramified subextension of K/Q p is M K/Q p. Since the restriction homomorphism Gal(M/Q p ) Gal(M K/Q p ) is surjective, the theorem follows. Theorem 2.8. The inertia group of the extension Q p (ζ n )/Q p is isomorphic to (Z/p e Z) where p e is the exact power of p dividing n. Proof By theorem 2.6 the inertia group of is isomorphic to the inertia group of Q(ζ n )/Q corresponding to p. Now let n = p e m. Then Gal(Q(ζ n )/Q) = (Z/p e Z) (Z/mZ) It is not hard to check that the fixed field of the subgroup isomorphic to (Z/mZ) is Q(ζ p e). Furthermore Q(ζ p e)/q is totally ramified with inertia group (Z/p e Z). Since Q(ζ m )/Q is unramified at p no further ramification occurs. Theorem 2.9. ( Hensel s Lemma) Let L be a local field, b be its maximal ideal, l be the residue field, f O L [x] be a monic polynomial, f be its restriction to l, and α l be such that f(α) = 0 and f (α) 0. Then there exists a root β of f in O L such that β = α (mod b).

4 4 NIZAMEDDIN H. ORDULU Proof Let β 0 O L be such that β 0 = α (mod b). Define β m = β m 1 f(β m 1) f (β m 1. It is an ) easy exercise to show that the sequence {β m } converges and the limit is a root of f. For a proof see [F-V] p. 36. Theorem If K/Q is unramified then K = Q Proof By a theorem of Minkowski dk ( π nn )s 4 n! where s is half the number of complex embeddings of K and n = [K : Q]. Using this one can show that if n > 1 then d K > 1 therefore there exists primes that are ramified. So if all primes are unramified, n = 1. Theorem Let K/Q be a Galois extension. The Galois group is generated by the inertia groups I p where p runs through all rational primes. Proof Let L be the fixed field of the group generated by I p s. Then L/Q is unramified so L = Q. The theorem follows. 3. Deriving the Global Theorem from the Local Case Theorem 3.1. The local Kronecker-Weber theorem implies the Global Kronecker-Weber theorem. Proof Assume that the local Kronecker-Weber theorem holds for all rational primes. Let K/Q be an abelian extension and p a rational prime that ramifies in K. Let b be a prime lying above p. Consider the localization K b /Q p. The Galois group is the decomposition group of b and hence the extension is abelian. By the local Kronecker-Weber theorem L b Q p (ζ np ) for some n p. Let p ep be the exact power of p dividing n p. Let n = p ep. p ramifies Claim 3.2. K Q(ζ n ) proof of the claim Let L = K(ζ n ). By the proof of theorem 2.7 we know that Q(ζ n )/Q is unramified outside n so the primes that ramify in L are the same as that of K. Let p be a prime that ramifies in L. Then by theorem 2.6 I p can be computed locally. The localization of L is L p =K b (ζ n ) Q p (ζ np, ζ n ) = Q p (ζ m ) where m is the least common multiple of n p and n. Now by theorem 2.8 the inertia groups of Q p (ζ m )/Q p and Q p (ζ n )/Q p are both isomorphic to

5 A SIMPLE PROOF OF KRONECKER-WEBER THEOREM 5 (Z/p e Z). Since Q p (ζ n ) L p Q p (ζ m ) by theorem 2.7, the inertia group of L p is (Z/p ep Z). Therefore I p = φ(p ep ). By theorem 2.11, Gal(L/Q) I p φ(n). p ramifies It follows that [L : Q] φ(n), but L already contains Q(ζ n ) and [Q(ζ n ) : Q] = φ(n). Therefore L = Q(ζ n ) from which it follows that K Q(ζ n ). Now let L/Q p be an abelian Galois extension. For the proof of the local Kronecker-Weber theorem we handle the following three cases seperately: The extension is unramified i.e. the maximal ideal of Q p remains prime in L. The extension is tamely ramified i.e. the ramification degree e is not divisible by p. The extension is wildly ramified i.e. the ramification degree e is divisible by p. 4. The Unramified Case We prove a stronger theorem from which the unramified case of the local Kronecker-Weber theorem follows. Theorem 4.1. Let L/K be an unramified, finite Galois extension where K and L are finite extensions of Q p.l = K(ζ n ) for some n with p n. Proof Assume that L/K is such an extension. Since e = 1 the inertia group is trivial and therefore the Galois group of L/K is isomorphic to the Galois group of the extension of the residue fields. Let α generate the extension of residue fields l/k. Since α is an element of a finite field with characteristic p, it is a root of unity with order coprime to p. Let n be the order of α. Now apply theorem 2.9 with f = x n 1 to obtain a root β O L of x n 1 such that β = α (mod b). Then [K(β) : K] [k(α) : k] but the latter has degree equal to [l : k] = [L : K] therefore L = K(β) = K(ζ n ). Now taking K = Q p gives us the desired result. We begin with two auxilary lemmata. 5. The Tamely Ramified Case

6 6 NIZAMEDDIN H. ORDULU Lemma 5.1. Let K and L be finite extensions of Q p and K the maximal ideal of O K. Suppose L/K is totally ramified of degree e with p e. Then there exists π K \ 2 K and a root α of x e π = 0 such that L = K(α). proof Let. denote the absolute value on C p. Let π 0 K \ 2 K and let β L be a uniformizing parameter so that β e = π 0. Then β e = π 0 u for some u U L (= the units of O L ) Now since f = 1 the extension of the residue fields is trivial, hence there exists u 0 U K such that u = u 0 (mod L ). Therefore u = u 0 + x with x L. Let π = π 0 u 0. Then β e = π 0 (u 0 + x) = π + π 0 x so β e π < π 0 = π. Let α 1, α 2,..., α e be the roots of f(x) = X e π. We claim that L = K(α i ) for some i. But Since α i e = π, α i = α j for all i, j. We have i 1 α i α j Max{ α i, α j } = α 1. α i α 1 = f (α 1 ) = eα e 1 1 = α 1 e 1. So α i α 1 = α 1, i 1. Since β α i = f(β) < π = i i we must have β α i < α 1 for some i. Without loss of generality assume that i = 1. Now let M be the Galois closure of the extension K(α 1, β)/k(β). Let σ Gal(M/K(β)). We have for i 1. But α i, β σ(α 1 ) = σ(β α 1 ) = β α 1 < α 1 = α i α 1 α 1 σ(α 1 ) Max{ α 1 β, β σ(α 1 ) } < α i α 1. It follows that σ(α 1 ) α i for i 1. So σ(α 1 ) = α 1. Since σ was arbitrary we have α 1 K(β) thus K(α 1 ) K(β) L. But f(x) is irreducible over K by Eisenstein criterion so [K(α 1 ) : K] = e = [L : K]. Therefore L = K(α 1 ). Lemma 5.2. Q p (( p) 1/(p 1) ) = Q p (ζ p )

7 A SIMPLE PROOF OF KRONECKER-WEBER THEOREM 7 Proof It is easy to prove that the maximal ideal of Q p (ζ p ) is given by (1 ζ p ). consider the polynomial g(x) = (X + 1)p 1 X = X p 1 + px p p Now Then so 0 = g(ζ p 1) (ζ p 1) p 1 + p (mod (ζ p 1) p ), u = (ζ p 1) p 1 1 (mod ζ p 1). p Let f(x) = X p 1 u then f(1) 0 (mod ζ p 1) and (ζ p 1) f (1). It follows from theorem 2.9 that there exists u 1 Q p (ζ p ) such that u p 1 1 = u. But then we have ( p) 1/(p 1) = ζ p 1 u 1 Q p (ζ p ) On the other hand X p 1 + p is irreducible over Q p by Eisenstein s criterion so Q p (( p) 1/(p 1) ) and Q p (ζ p ) have the same degree over Q p. Therefore Q p (( p) 1/(p 1) ) = Q p (ζ p ). Now let L/Q p be a tamely ramified abelian extension. Let K/Q p be the maximal unramified subextension. Then K Q p (ζ n ) for some n by the previous section. L/K is totally ramified with degree p e. By lemma 5.1 L = K(π 1/e ) for some π of order 1 in K. Since K/Q p is unramified, p is of order 1 in K, so π = up for some unit u K. Since u is a unit and p e the discriminant of f(x) = X e u is not divisible by p, hence K(u 1/e )/K is unramified. By theorem 4.1 K(u 1/e ) K(ζ M ) Q p (ζ Mn ) for some M. Let T be the compositum of the fields Q p (ζ Mn ) and L. By theorem 2.2, T/Q p is abelian. Since u 1/e, π 1/e T ( p) 1/e T. It follows that Q p (( p) 1/e )/Q p is Galois since it is a subextension of the abelian extension T/Q p. Therefore ζ e Q p (( p) 1/e ). Since Q p (( p) 1/e ) is totally ramified, so is the subextension Q p (ζ e )/Q p. But p e, so the latter extension is trivial and ζ e Q p. Therefore e (p 1). Now by lemma 5.2, Therefore This finishes the tamely ramified case. Q p (( p) 1/e ) Q p (ζ p ). L = K(π 1/e ) = K(u 1/e, ( p) 1/e ) Q p (ζ Mnp ).

8 8 NIZAMEDDIN H. ORDULU 6. The Wildly Ramified Case This part of the proof requires knowledge of Kummer theory. We briefly sketch the proof for details see [W] p Assume that p is an odd prime. First of all note that we may assume by structure theorem for abelian groups and theorem 2.3, that the extension L/Q p is cyclic, totally ramified of degree p m for some m. Now let K u /Q p be an unramified cyclic extension of degree p m and let K r /Q p be a totally ramified extension of degree p m. (K u can be obtained by taking the extension F/F p of degree p m and lifting the minimal polynomial of its primitive element to Z p [X]. The root of this polynomial will generate an unramified extension of degree p m. K r can be taken to be the fixed field of the subgroup isomorphic to (Z/pZ) in the extension Q p (ζ p m)/q p.) By the unramified case of the thorem we know that K u Q p (ζ n ) for some n. Since K r K u = Q p, by theorem 2.2, If L K r K u then Gal(K r K u /Q p ) = (Z/p m Z) 2. Gal(K(ζ n, ζ p m+1)/q p ) = (Z/p m Z) 2 Z/p m Z for some m > 0. This group has (Z/pZ) 3 as a quotient, so there is a field N such that Gal(N/Q p ) = (Z/pZ) 3. Following lemma shows that this is impossible. Lemma 6.1. Let p be an odd prime. There is no extension N/Q p such that Gal(N/Q p ) = (Z/pZ) 3. Before proving the above lemma we quote the following lemma without proof. Interested reader can find the proof in [W] p Lemma 6.2. Let F be a field of characteristic p, let M = F (ζ p ), and let L = M(a 1/p ) for some a M. Define the character ω : Gal(M/F ) F p by σ(ζ p ) = ζp ω(σ). Then for all σ Gal(M/F ). L/F is abelian σ(a) = a ω(σ) (mod (M ) p ) Proof of 6.1 Assume that there exists such an N, then N(ζ p )/Q p is abelian and Gal(N(ζ p )/Q p (ζ p )) = (Z/pZ) 3. This is a Kummer extension so there is a corresponding subgroup B Q p (ζ p ) /(Q p (ζ p ) ) p with B = (Z/pZ) 3 and Q p (ζ p )(B 1/p ) = N(ζ p ). Let a B and L = Q p (ζ p, a 1/p ) N(ζ p ). Since L/Q p is abelian, by lemma 6.2, σ(a) = a ω(σ) (mod(q p (ζ p ) ) p ), σ Gal(Q p (ζ p )/Q p ).

9 A SIMPLE PROOF OF KRONECKER-WEBER THEOREM 9 Let v be the valuation on Q p (ζ p ) such that v(ζ p 1) = 1. Then v(a) = v(σ(a)) = ω(σ)v(a) (mod p), for al σ. Now if σ id the above equality gives v(a) = 0 (mod p).it is easy to verify that Q p (ζ p ) = (ζ p 1) Z W p 1 U 1 where W p 1 are the roots of unity in Q p and U 1 = {u = 1 (mod ζ p 1)}. Since p v(a) and W p 1 s elements are already pth powers, a is equivalent to an element in U 1. So assume a U 1. We can also assume B U 1 /U p 1, and Gal(Q p (ζ p )/Q p ) acts via ω. We claim that U p 1 = {u = 1 (mod π p+1 )}. Let π = 1 ζ p. Now if u U 1 then u = 1 + πx. By looking at the binomial expansion one can show that u p = 1 (mod π p+1 ). Conversely if u 2 = 1 (mod π p+1 ) then the binomial series for (1 + u 2 1) 1/p converges. This proves the claim. Let u B. Let u = 1+bπ +. Since ζ p = 1+π we have ζ b p = 1+bπ +. Thus u = ζ b pu 1 with u 1 = 1 (mod π 2 ). Since σ(u) = u ωσ (mod U p 1 ) substituting u = u 1 ζ b p yields σ(u 1 ) = u ωσ 1 (mod U p 1 ). Write with c Z, p c, and d 2. Note that u 1 = 1 + cπ d + σ(π) π = ζω(σ) p 1 ζ p 1 = ζωσ 1 p = ω(σ) (modπ). So (σ(π))/π = ω(σ) (mod π).we have but σ(u 1 ) = 1 + cω(σ) d π d + u ω(σ) 1 = 1 + cω(σ)π d + Since σ(u 1 ) = u ωσ 1 (mod U p 1 ) and U p 1 = {u = 1 (mod π p+1 )}, we have σ(u 1 ) = u ω(σ) 1 (mod π p+1 ). This means that either d p + 1 or d = 1 (mod p 1). The former means that u 1 U p 1 and the latter means that d = p. Clearly 1 + π p generates modulo U p 1 the subgroup of u 1 = 1 (mod π p ). We therefore obtained B ζ p, 1 + π p where x, y denotes the group generated by x and y. Since B = (Z/pZ) 3, we have a contradiction.

10 10 NIZAMEDDIN H. ORDULU For p = 2 one has to make a more careful analysis so we shall omit it here. For the proof of this case see [W] p References [S1] William Stein, A Brief Introduction to Classical and Adelic Algebraic Number Theory, Course Notes (2004). [S2] William Stein, Introduction to Algebraic Number Theory, Course Notes (2005) [G] Fernando Q. Gouvea, p-adic numbers, (1997). [W] Lawrence C. Washington, Introduction to Cyclotomic Fields, (1997) [F-V] I. B. Fesenko, S. V. Vostokov Local Fields and their extensions, (2002) address:

### 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

### Galois Theory TCU Graduate Student Seminar George Gilbert October 2015

Galois Theory TCU Graduate Student Seminar George Gilbert October 201 The coefficients of a polynomial are symmetric functions of the roots {α i }: fx) = x n s 1 x n 1 + s 2 x n 2 + + 1) n s n, where s

### 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

### 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

### 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,

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

### 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

### 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

### 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

### 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

### ALGEBRA HW 9 CLAY SHONKWILER

ALGEBRA HW 9 CLAY SHONKWILER 1 Let F = Z/pZ, let L = F (x, y) and let K = F (x p, y p ). Show that L is a finite field extension of K, but that there are infinitely many fields between K and L. Is L =

### 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

### Class Field Theory. Travis Dirle. December 4, 2016

Class Field Theory 2 Class Field Theory Travis Dirle December 4, 2016 2 Contents 1 Global Class Field Theory 1 1.1 Ray Class Groups......................... 1 1.2 The Idèlic Theory.........................

### 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

### 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

### 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

### GALOIS THEORY. Contents

GALOIS THEORY MARIUS VAN DER PUT & JAAP TOP Contents 1. Basic definitions 1 1.1. Exercises 2 2. Solving polynomial equations 2 2.1. Exercises 4 3. Galois extensions and examples 4 3.1. Exercises. 6 4.

### 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

### 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

### Galois Theory of Cyclotomic Extensions

Galois Theory of Cyclotomic Extensions Winter School 2014, IISER Bhopal Romie Banerjee, Prahlad Vaidyanathan I. Introduction 1. Course Description The goal of the course is to provide an introduction to

### 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.

### MATH 101A: ALGEBRA I, PART D: GALOIS THEORY 11

MATH 101A: ALGEBRA I, PART D: GALOIS THEORY 11 3. Examples I did some examples and explained the theory at the same time. 3.1. roots of unity. Let L = Q(ζ) where ζ = e 2πi/5 is a primitive 5th root of

### 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

### 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

### 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

### 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

### Thus, the integral closure A i of A in F i is a finitely generated (and torsion-free) A-module. It is not a priori clear if the A i s are locally

Math 248A. Discriminants and étale algebras Let A be a noetherian domain with fraction field F. Let B be an A-algebra that is finitely generated and torsion-free as an A-module with B also locally free

### 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

### 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

### THE JOHNS HOPKINS UNIVERSITY Faculty of Arts and Sciences FINAL EXAM - SPRING SESSION ADVANCED ALGEBRA II.

THE JOHNS HOPKINS UNIVERSITY Faculty of Arts and Sciences FINAL EXAM - SPRING SESSION 2006 110.402 - ADVANCED ALGEBRA II. Examiner: Professor C. Consani Duration: 3 HOURS (9am-12:00pm), May 15, 2006. No

### Contents Lecture 1 2 Norms 2 Completeness 4 p-adic expansions 5 Exercises to lecture 1 6 Lecture 2 7 Completions 7 p-adic integers 7 Extensions of Q

p-adic ANALYSIS, p-adic ARITHMETIC Contents Lecture 1 2 Norms 2 Completeness 4 p-adic expansions 5 Exercises to lecture 1 6 Lecture 2 7 Completions 7 p-adic integers 7 Extensions of Q p 10 Exercises to

### CYCLOTOMIC EXTENSIONS

CYCLOTOMIC EXTENSIONS KEITH CONRAD 1. Introduction For a positive integer n, an nth root of unity in a field is a solution to z n = 1, or equivalently is a root of T n 1. There are at most n different

### 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

### MAT 535 Problem Set 5 Solutions

Final Exam, Tues 5/11, :15pm-4:45pm Spring 010 MAT 535 Problem Set 5 Solutions Selected Problems (1) Exercise 9, p 617 Determine the Galois group of the splitting field E over F = Q of the polynomial f(x)

### Math 414 Answers for Homework 7

Math 414 Answers for Homework 7 1. Suppose that K is a field of characteristic zero, and p(x) K[x] an irreducible polynomial of degree d over K. Let α 1, α,..., α d be the roots of p(x), and L = K(α 1,...,α

### Lecture 7: Etale Fundamental Group - Examples

Lecture 7: Etale Fundamental Group - Examples October 15, 2014 In this lecture our only goal is to give lots of examples of etale fundamental groups so that the reader gets some feel for them. Some of

### Solutions for Problem Set 6

Solutions for Problem Set 6 A: Find all subfields of Q(ζ 8 ). SOLUTION. All subfields of K must automatically contain Q. Thus, this problem concerns the intermediate fields for the extension K/Q. In a

### CLASSIFYING EXTENSIONS OF THE FIELD OF FORMAL LAURENT SERIES OVER F p

CLASSIFYING EXTENSIONS OF THE FIELD OF FORMAL LAURENT SERIES OVER F p ALFEEN HASMANI, LINDSEY HILTNER, ANGELA KRAFT, AND DANIEL SCOFIELD Abstract. In previous works, Jones and Roberts have studied finite

### 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

### 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

### 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

### 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

### 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

### Homework 4 Algebra. Joshua Ruiter. February 21, 2018

Homework 4 Algebra Joshua Ruiter February 21, 2018 Chapter V Proposition 0.1 (Exercise 20a). Let F L be a field extension and let x L be transcendental over F. Let K F be an intermediate field satisfying

### 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

### ON 3-CLASS GROUPS OF CERTAIN PURE CUBIC FIELDS. Frank Gerth III

Bull. Austral. ath. Soc. Vol. 72 (2005) [471 476] 11r16, 11r29 ON 3-CLASS GROUPS OF CERTAIN PURE CUBIC FIELDS Frank Gerth III Recently Calegari and Emerton made a conjecture about the 3-class groups of

### HASSE ARF PROPERTY AND ABELIAN EXTENSIONS. Ivan B. Fesenko

Math. Nachrichten vol. 174 (1995), 81 87 HASSE ARF PROPERTY AND ABELIAN EXTENSIONS Ivan B. Fesenko Let F be a complete (or Henselian) discrete valuation field with a perfect residue field of characteristic

### 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

### Fields and Galois Theory. Below are some results dealing with fields, up to and including the fundamental theorem of Galois theory.

Fields and Galois Theory Below are some results dealing with fields, up to and including the fundamental theorem of Galois theory. This should be a reasonably logical ordering, so that a result here should

### 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

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.

### Solutions of exercise sheet 6

D-MATH Algebra I HS 14 Prof. Emmanuel Kowalski Solutions of exercise sheet 6 1. (Irreducibility of the cyclotomic polynomial) Let n be a positive integer, and P Z[X] a monic irreducible factor of X n 1

### 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

### 22. Galois theory. G = Gal(L/k) = Aut(L/k) [L : K] = H. Gal(K/k) G/H

22. Galois theory 22.1 Field extensions, imbeddings, automorphisms 22.2 Separable field extensions 22.3 Primitive elements 22.4 Normal field extensions 22.5 The main theorem 22.6 Conjugates, trace, norm

### 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

### 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

### 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

### 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

### Extension fields II. Sergei Silvestrov. Spring term 2011, Lecture 13

Extension fields II Sergei Silvestrov Spring term 2011, Lecture 13 Abstract Contents of the lecture. Algebraic extensions. Finite fields. Automorphisms of fields. The isomorphism extension theorem. Splitting

### 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

### 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)

### 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

### IDEAL CLASS GROUPS OF CYCLOTOMIC NUMBER FIELDS I

IDEA CASS GROUPS OF CYCOTOMIC NUMBER FIEDS I FRANZ EMMERMEYER Abstract. Following Hasse s example, various authors have been deriving divisibility properties of minus class numbers of cyclotomic fields

### arxiv: v1 [math.gr] 3 Feb 2019

Galois groups of symmetric sextic trinomials arxiv:1902.00965v1 [math.gr] Feb 2019 Alberto Cavallo Max Planck Institute for Mathematics, Bonn 5111, Germany cavallo@mpim-bonn.mpg.de Abstract We compute

### Algebra SEP Solutions

Algebra SEP Solutions 17 July 2017 1. (January 2017 problem 1) For example: (a) G = Z/4Z, N = Z/2Z. More generally, G = Z/p n Z, N = Z/pZ, p any prime number, n 2. Also G = Z, N = nz for any n 2, since

### Galois Representations

9 Galois Representations This book has explained the idea that all elliptic curves over Q arise from modular forms. Chapters 1 and introduced elliptic curves and modular curves as Riemann surfaces, and

### Keywords and phrases: Fundamental theorem of algebra, constructible

Lecture 16 : Applications and Illustrations of the FTGT Objectives (1) Fundamental theorem of algebra via FTGT. (2) Gauss criterion for constructible regular polygons. (3) Symmetric rational functions.

### Galois Theory, summary

Galois Theory, summary Chapter 11 11.1. UFD, definition. Any two elements have gcd 11.2 PID. Every PID is a UFD. There are UFD s which are not PID s (example F [x, y]). 11.3 ED. Every ED is a PID (and

### 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

### CONSTRUCTION OF THE HILBERT CLASS FIELD OF SOME IMAGINARY QUADRATIC FIELDS. Jangheon Oh

Korean J. Math. 26 (2018), No. 2, pp. 293 297 https://doi.org/10.11568/kjm.2018.26.2.293 CONSTRUCTION OF THE HILBERT CLASS FIELD OF SOME IMAGINARY QUADRATIC FIELDS Jangheon Oh Abstract. In the paper [4],

### 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

### GALOIS THEORY AT WORK

GALOIS THEORY AT WORK 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 determined by their

### Chapter 4. Fields and Galois Theory

Chapter 4 Fields and Galois Theory 63 64 CHAPTER 4. FIELDS AND GALOIS THEORY 4.1 Field Extensions 4.1.1 K[u] and K(u) Def. A field F is an extension field of a field K if F K. Obviously, F K = 1 F = 1

### 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

### NOTES ON FINITE FIELDS

NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining

### 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)

### 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

### PMATH 441/641 ALGEBRAIC NUMBER THEORY

PMATH 441/641 ALGEBRAIC NUMBER THEORY 1. Lecture: Wednesday, January 5, 000 NO TEXT References: Number Fields (Marcus) Algebraic Number Theory (Lang) (Stewart & Hill) Marks: Final Exam 65% Midterm 5% Assignments

### Algebra Qualifying Exam Solutions. Thomas Goller

Algebra Qualifying Exam Solutions Thomas Goller September 4, 2 Contents Spring 2 2 2 Fall 2 8 3 Spring 2 3 4 Fall 29 7 5 Spring 29 2 6 Fall 28 25 Chapter Spring 2. The claim as stated is false. The identity

### 1 Number Fields Introduction Algebraic Numbers Algebraic Integers Algebraic Integers Modules over Z...

Contents 1 Number Fields 3 1.1 Introduction............................ 3 1.2 Algebraic Numbers........................ 5 2 Algebraic Integers 7 2.1 Algebraic Integers......................... 7 2.2 Modules

### Algebraic Number Theory

Algebraic Number Theory Andrew Kobin Spring 2016 Contents Contents Contents 0 Introduction 1 0.1 Attempting Fermat s Last Theorem....................... 1 1 Algebraic Number Fields 4 1.1 Integral Extensions

### HISTORY OF CLASS FIELD THEORY

HISTORY OF CLASS FIELD THEORY KEITH CONRAD 1. Introduction Class field theory is the description of abelian extensions of global fields and local fields. The label class field refers to a field extension

### CYCLOTOMIC EXTENSIONS AND QUADRATIC RECIPROCITY

CYCLOTOMIC EXTENSIONS AND QUADRATIC RECIPROCITY ALEXANDER BERTOLONI MELI Abstract. We develop some of the basic theory of algebraic number fields and cyclotomic extensions and use this to give a proof

### Galois Representations

Galois Representations Samir Siksek 12 July 2016 Representations of Elliptic Curves Crash Course E/Q elliptic curve; G Q = Gal(Q/Q); p prime. Fact: There is a τ H such that E(C) = C Z + τz = R Z R Z. Easy

### Solutions of exercise sheet 11

D-MATH Algebra I HS 14 Prof Emmanuel Kowalski Solutions of exercise sheet 11 The content of the marked exercises (*) should be known for the exam 1 For the following values of α C, find the minimal polynomial

### Number Theory Fall 2016 Problem Set #6

18.785 Number Theory Fall 2016 Problem Set #6 Description These problems are related to the material covered in Lectures 10-12. Your solutions are to be written up in latex (you can use the latex source

### ALGEBRA PH.D. QUALIFYING EXAM September 27, 2008

ALGEBRA PH.D. QUALIFYING EXAM September 27, 2008 A passing paper consists of four problems solved completely plus significant progress on two other problems; moreover, the set of problems solved completely

### Section X.55. Cyclotomic Extensions

X.55 Cyclotomic Extensions 1 Section X.55. Cyclotomic Extensions Note. In this section we return to a consideration of roots of unity and consider again the cyclic group of roots of unity as encountered

### On attaching coordinates of Gaussian prime torsion points of y 2 = x 3 + x to Q(i)

On attaching coordinates of Gaussian prime torsion points of y 2 = x 3 + x to Q(i) Gordan Savin and David Quarfoot March 29, 2010 1 Background One of the natural questions that arises in the study of abstract

### 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

### 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

### Pacific Journal of Mathematics

Pacific Journal of Mathematics MOD p REPRESENTATIONS ON ELLIPTIC CURVES FRANK CALEGARI Volume 225 No. 1 May 2006 PACIFIC JOURNAL OF MATHEMATICS Vol. 225, No. 1, 2006 MOD p REPRESENTATIONS ON ELLIPTIC

### 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

### ALGEBRA PH.D. QUALIFYING EXAM SOLUTIONS October 20, 2011

ALGEBRA PH.D. QUALIFYING EXAM SOLUTIONS October 20, 2011 A passing paper consists of four problems solved completely plus significant progress on two other problems; moreover, the set of problems solved

### 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

### arxiv: v1 [math.nt] 2 Jul 2009

About certain prime numbers Diana Savin Ovidius University, Constanţa, Romania arxiv:0907.0315v1 [math.nt] 2 Jul 2009 ABSTRACT We give a necessary condition for the existence of solutions of the Diophantine

### 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.

### Ideal class groups of cyclotomic number fields I

ACTA ARITHMETICA XXII.4 (1995) Ideal class groups of cyclotomic number fields I by Franz emmermeyer (Heidelberg) 1. Notation. et K be number fields; we will use the following notation: O K is the ring