A Simple Proof of the Kronecker-Weber Theorem [bachelor’s thesis]

43 Pages·2012·0.333 MB·English

by  Daniel Lupp

BACHELOR THESIS
A Simple Proof of the
Kronecker-Weber Theorem
by
Daniel Lupp
supervised by
Prof. Dr. N. Scheithauer
FB04 Mathematik
TU Darmstadt
November 2011

Eidesstattliche Erkl¨arung
Ich versichere, dass ich meine Bachelorarbeit ohne Hilfe Dritter und ohne Benutzung
anderer als der angegebenen Quellen und Hilfsmittel angefertigt und die den benutzten
Quellen wo¨rtlich oder inhaltlich entnommenen Stellen als solche kenntlich gemacht habe.
Diese Arbeit hat in gleicher oder a¨hnlicher Form noch keiner Pru¨fungsbeho¨rde vorgele-
gen.
Darmstadt, den Unterschrift:
3

Contents
Introduction 7
1 Review of Facts 9
1.1 Some Results from Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Field Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Galois Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Norm, Trace, and Discriminant . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Cyclotomic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Some Results from Algebraic Number Theory 17
2.1 Dedekind Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Ramification Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Algebraic Number Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4 Ramification in Cyclotomic Extensions . . . . . . . . . . . . . . . . . . . 31
3 The Kronecker-Weber Theorem 35
5

Introduction
Averyimportantresultinalgebraicnumbertheory, theKronecker-Webertheoremstates
that every finite abelian Galois extension of Q is contained in a cyclotomic field. This is
quite a remarkable statement: It is not very difficult to prove that cyclotomic extensions
ofQarefiniteandabelian. Atfirstglancethiswouldseemtobearathersmallclassofthe
finite abelian extensions of Q. However, Kronecker-Weber states that with cyclotomic
extensions we have, in fact, captured all of these extensions. In addition, it has some
interesting corollaries. For example, as a consequence of Kronecker-Weber one knows
that every algebraic integer in a finite abelian extension of Q can be described as a
polynomial in Z[ξ], where ξ is a suitable root of unity.
Kronecker-Weber is usually proved using very heavy machinery from class field theory.
However, there is a quite simple proof, as published by Greenberg [Gre74] and based on
a proof by Hilbert, which uses only basic notions from algebraic number theory. The
necessary facts from field extension theory, in particular Galois theory and cyclotomic
fields, are reviewed, without proof, in Chapter 1. For a more detailed look at these
topics, see [Bos09]. In Chapter 2 we deal with important results from algebraic number
theory, in particular the concept of Dedekind domains and ramification of prime ideals,
with an emphasis on ramification in cyclotomic extensions. This gives us the necessary
tools for the proof of the Kronecker-Weber theorem, which is presented in Chapter 3.
7

1 Review of Facts
1.1 Some Results from Algebra
This section deals with general results concerning rings with [Neu07] as a primary refer-
ence. Throughout this thesis, rings will be understood to be commutative unitary rings.
We shall begin with the following definitions concerning integrality.
Definition 1.1.1. Let A,B be rings such that A ⊂ B. An element b ∈ B is said to be
integral over A, if it satisfies a monic equation
xs +r xs−1 +...+r = 0, r ∈ A.
s−1 0 i
¯
The integral closure A of A in B is defined as the set of all b ∈ B which are integral
¯
over A. If A is its own integral closure in B, i.e., A = A, then A is said to be integrally
closed.
¯
One can show that the integral closure A of a ring A in B is integrally closed in B.
¯
Furthermore, A is a ring [Neu07, pp. 6-7].
We shall now give a proof of the Chinese remainder theorem for rings, following [IR90,
pp. 181-182]. Furthermore, we will briefly introduce local rings and discrete valuation
rings.
Theorem 1.1.2 (Chinese Remainder Theorem). Let R be a commutative ring with
identity. For ideals A ,...,A with A +A = R, i (cid:54)= j, let A = A A ···A . Then we
1 n i j 1 2 n
have the following isomorphism:
∼
R/A = R/A ⊕R/A ⊕···⊕A .
1 2 n
Proof. Let φ denote the natural projection from R to R/A . We will show that the map
i i
φ : R → R/A ⊕R/A ⊕···⊕A , φ(γ) = (φ (γ),φ (γ),...,φ (γ)) is surjective and has
1 2 n 1 2 n
A as its kernel.
For the surjectivity claim, we will show that for all γ ,...,γ ∈ R the set of congruences
1 n
X ≡ γ mod A , i = 1,...,n is simultaneously solvable. Since A +A = R for i (cid:54)= j. we
i i i j
see that (A +A )(A +A )···(A +A ) = R. By expansion of this product we find that
1 2 1 3 1 n
all summands except the last contain A as a factor, i.e., are contained in A . Hence
1 1
A + A ···A = R. Then there exist elements v ∈ A and u ∈ A ···A such that
1 2 n 1 1 1 2 n
v + u = 1 and thus u ≡ 1 mod A and u ≡ 0 mod A , i (cid:54)= 1. Repeating the same
1 1 1 1 1 i
argumentforallA , wecanchooseu suchthatu ≡ 1 mod A andu ≡ 0 mod A , i (cid:54)= j.
i i i i i j
Then the element x = γ u +...+γ u is a solution for the set of congruences, implying
1 1 n n
that φ is surjective.
9
To prove tha A is the kernel of φ, it suffices to show that A = A ∩ ... ∩ A , since
1 n
ker(φ) = A ∩...∩A . We shall proceed by induction over n. For n = 2 we must only
1 n
show that A ∩A ⊂ A A , since the other inclusion is trivial. Since A +A = R we
1 2 1 2 1 2
can find a ∈ A and a ∈ A , such that a +a = 1. Then for a ∈ A ∩A we have by
1 1 2 2 1 2 1 2
distributivity that a = aa +aa ∈ A A and hence A ∩A ⊂ A A . For n > 2 we have
1 2 1 2 1 2 1 2
A ∩A ∩...∩A = A ∩A A ···A by induction hypothesis. We have shown above
1 2 n 1 2 3 n
that A +A A ···A = R and so we find A ∩A A ···A = A A ···A , proving that
1 2 3 n 1 2 3 n 1 2 n
A is the kernel of φ. Thus φ induces the desired isomorphism.
A unitary ring A is called a local ring, if it has a unique maximal ideal p. For an
element a ∈ A not in p, we know that the principal ideal (a) is not contained in p and
therefore is not contained in a maximal ideal. This implies (a) = A and a is a unit, i.e.,
we have A∗ = A\p.
The following definition deals with the special case that A is a principal ideal domain.
Definition 1.1.3. A principal ideal domain A with a unique maximal ideal p (cid:54)= 0 is
called a discrete valuation ring.
For a discrete valuation ring A let p = (π) be its maximal ideal. Since A is a local
ring, each element x ∈ A with x (cid:54)∈ p is a unit. Hence π and its associated elements are
the only prime elements in A. Therefore every nonzero element a ∈ A can be written as
a = απn for α ∈ A∗ and n ≥ 0. By allowing the exponents n to be integers instead of
just natural numbers, we are able to capture all elements of the field of fractions K, i.e.,
for 0 (cid:54)= a ∈ K we have a = απn, for α ∈ A∗ and n ∈ Z. This gives rise to the notion of
the valuation v(a) of a. It is defined as the exponent n of the above factorization, i.e.,
(a) = pv(a). Thevaluationisamapv : K∗ → Zandcanbeextendedtov : K → Z∪{∞}
by defining v(0) = ∞. Then it has the following properties:
1. v(ab) = v(a)+v(b)
2. v(a+b) ≥ min(v(a),v(b))
1.2 Field Extensions
We will now review various types of field extensions.
Definition 1.2.1. AfieldLiscalledafield extension of K,ifK isasubfieldofL,denoted
as the field extension L/K. The degree of the field extension, denoted by [L : K], is
the dimension of L as a K vector space. A field extension L/K is called a finite field
extension, if its degree is finite.
A field extension L/K where every element in L is algebraic over K, i.e., is the root of
a polynomial with coefficients in K, is called an algebraic field extension. In particular,
every finite field extension is algebraic.
A normal field extension L/K is an algebraic field extension in which, for every a ∈ L,
its minimal polynomial m in K[X] splits into linear factors in L. For example, the
a,K
10