The Kronecker-Weber theorem [Bachelor thesis]

34 Pages·2016·0.461 MB·English

by  Romain Branchereau

THE KRONECKER-WEBER THEOREM
a Bachelor Thesis
written by supervised by
Romain Branchereau Prof. Dr. Richard Pink
and Alexandre Puttick
Abstract
We present an elementary proof of the Kronecker-Weber Theorem and introduce the
necessary mathematical background.
May 2016
CONTENTS 2
Contents
Introduction 3
1 Ramification Theory 5
1.1 The ring of integers and factorization of primes . . . . . . . . . . . . . . . . . . . 5
1.2 Ramification of primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Ramification of primes in Galois extensions . . . . . . . . . . . . . . . . . . . . . 7
1.4 Decomposition and Inertia groups . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Compositum of fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Valuations 13
2.1 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Localization of ring of integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Higher ramification groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Cyclotomic fields 19
3.1 Recollections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 The Galois group of cyclotomic field extensions . . . . . . . . . . . . . . . . . . . 20
3.3 Ramification in cyclotomic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4 The quadratic Gauss sum 22
5 Proof of the Kronecker-Weber Theorem 25
5.1 A few Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.2 Proof for cyclic p-power extensions unramified outside p . . . . . . . . . . . . . . 27
5.2.1 The case p=2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2.2 The case p>2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.3 Proof for cyclic p-power extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.4 Proof for any abelian extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
A Fundamental Theorem of Galois Theory 33
CONTENTS 3
Introduction
The aim of this Bachelor Thesis is to present an elementary proof due to Greenberg [Gre74] of
the following theorem.
Theorem (Kronecker-Weber). Every finite abelian extension K|Q is cyclotomic.
Inotherwords,thismeansthatifaGaloisextensionK|QhasafiniteabelianGaloisgroup,then
K is contained in some cyclotomic field.
The Theorem was first stated by Leopold Kronecker in 1853, but his proof was incomplete.
In 1886, Heinrich Weber presented a new proof that was still incomplete. David Hilbert finally
proved it in 1896 using different techniques, and considered the generalization of this theorem
concerning abelian Galois extension of general number fields instead of Q. This generalization is
known as Hilbert’s 12th problem.
In modern literature, the proofs of the Kronecker-Weber Theorem are usually based on class
fieldtheory. Theproofthatwepresentinsection5usesmoreelementaryconceptsfromalgebraic
number theory, which will be introduced in sections 1 to 4. Standard results from Algebra, and
inparticularfromGaloisTheorywillbeassumed. However,theFundamentalTheoremofGalois
Theory is stated in the appendix A.1, since we use it several times. The classical results that
canbefoundinmostbooksaboutalgebraicnumbertheorywillusuallybestatedwithoutproof,
whereas the results that are more specific to the proof of the Kronecker-Weber Theorem will be
proven.
I would like to thank my supervisors, Professor Richard Pink and Alexandre Puttick, for
their helpful comments on drafts of this paper.
CONTENTS 4
Notation and terminology
The rings we consider are always commutative with unity, and the fields are always number
fields. When we say "abelian/cyclic extensions" we mean "Galois extension with abelian/cyclic
Galoisgroup". Severalproofsaredividedintoclaims. Inthiscasetheendoftheproofofaclaim
is indicated by (cid:4) whereas the end of the whole proof is indicated by (cid:3).
R× Multiplicative group of units of a ring R
U(n) The multiplicative group (Z/nZ)×
[L:K] Degree of a field extension L|K
(G:H) Index of a subgroup H in G
H ≤G H is a subgroup of G
H (cid:69)G H is a normal subgroup of G
Quot(R) The field of fraction of R
G Galois group Gal(K|Q)
K
1 RAMIFICATION THEORY 5
1 Ramification Theory
1.1 The ring of integers and factorization of primes
Let R ⊆ S be a ring extension. An element s ∈ S is integral over R if it is a root of a monic
polynomial with coefficients in R, i.e. there exist n≥1 and elements r ,...,r ∈R such that
1 n
sn+r sn−1+···+r s+r =0.
1 n−1 0
We will assume the following Proposition:
Proposition 1.1. Let R ⊂ S be a ring extension and s ,...,s ∈ S. The following are
1 m
equivalent:
1. s ,...,s are integral over R,
1 m
2. R[s ,...,s ] is a finitely generated R-module.
1 m
Proof. See [AM69, Proposition 5.1].
Corollary 1.2. Let R:={s∈S |sis integral overR}. Then R is a subring of S containing R.
Proof. Every element r ∈R is integral over R, since it is a root of the monic polynomial x−r.
It follows that R ⊆ R. Let x,y ∈ R. Then R[x,y] is finitely generated as an R-module by
Proposition 1.1. On the other hand, we know that R[x,y,xy]=R[x,y]. Thus xy ∈R according
to the same Proposition. Similarly for x+y.
The ring R is called the integral closure of R in S. If R = R we say that R is integrally closed
in S. If R=S we say that S is integral over R.
Definition 1.3. LetK beanumberfield,i.e. afinitedegreeextensionofQ. Theintegralclosure
of Z in K is called the ring of integers of K and is denoted by O . Equivalently, an element x
K
is in O if there exist n≥1 and a ,...,a ∈Z such that
K 0 n−1
xn+a xn−1+···+a x+a =0.
n−1 1 0
If K = Q, the ring of integers is OQ = Z. A very useful property of Z is that it is a unique
factorization domain, that allows us to decompose any nonzero integer into a product of primes
in a unique way. Unfortunately, this property does not hold for an arbitrary ring of integers.
√ √
For example, one can show that the ring of integers of Q( −5) is Z[ −5] which is not a unique
factorization domain, since we have
√ √
2·3=6=(1+ −5)·(1− −5).
Remark 1.4. Note that in general it is not true that the ring of integers of Q(α) is Z[α].
Definition1.5. ADedekinddomainDisanintegraldomainsatisfyingthefollowingproperties:
1. D is noetherian,
2. D is integrally closed,
1 RAMIFICATION THEORY 6
3. every nonzero prime ideal in D is maximal.
ThefundamentalpropertyofDedekindDomainsisthattheyallowauniquefactorizationofnon
zero ideals into prime ideals.
Theorem 1.6. Let D be a Dedekind domain and a be a nonzero proper ideal of D. Then we
have a unique decomposition
a=pe1···pek
1 k
where p ,··· ,p are distinct nonzero prime ideals and e ,··· ,e are positive integers.
1 k 1 k
Proof. See [Neu99, Theorem 3.3 in Chapter 1].
Theorem 1.7. For any number field K, the ring of integers O is a Dedekind domain.
K
Proof. See [Neu99, Theorem 3.1 in Chapter 1].
By Theorem 1.6, every nonzero ideal of O can be factored into prime ideals. This is a the
K
generalization of the factorization into prime numbers that we have in Z. The field of rationals
Q is constructed by taking all the fractions of the integers Z, hence Q is the field of fractions of
Z. The following Proposition shows that this is also true for an arbitrary number field.
Proposition 1.8. Let K be a number field. Then K is the field of fractions of O .
K
Proof. Lety ∈K,thenyisalgebraic,i.e. itsatisfiesarelationa yn+a yn−1+···a y+a =0,
n n−1 1 0
where a ∈Z and a 6=0. Let x:=ya . Multiplying both sides by an−1, we get
i n n n
xn+a xn−1+···a an−2x+a an−1 =0.
n−1 1 n 0 n
Thus x ∈ O and y = x ∈ Quot(O ). On the other hand, we have O ⊆ K. Thus
K an K K
Quot(O )⊆Quot(K)=K, since K is a field.
K
1.2 Ramification of primes
Let K be a number field and n = [K : Q]. Let p be a nonzero prime ideal in O . Then p∩Z
K
is a prime ideal in Z, generated by some prime p. So (p) = p∩Z and we say that p lies over p
or that p lies under p. Let pO be the ideal generated by p in O . Although the ideal (p) is
K K
prime in Z, the ideal pO is not necessarily prime in O .
K K
√ √
Example 1.9. Consider K =Q( −5), with ring of integers Z[ −5]. In Z the ideal generated
√
by p=29 is prime, but in Z[ −5] we have the decomposition
√ √
29=(3+2 −5)·(3−2 −5),
√
hence the ideal generated by 29 is not prime in Z[ −5].
SinceO isaDedekinddomain,weknowfromTheorem1.6thatwehaveauniquefactorization
K
into prime ideals pOK = pe11···pegg, where the prime ideals pi are exactly the ideals lying over
p. We will also say that p divides p and use the notation p |p.
i i
Definition 1.10. The exponent e is called the ramification index of p over p.
i i
1 RAMIFICATION THEORY 7
Wealsowritee (p |p)fortheramificationindexifitisnotclearwhichidealsareconcerned. Since
i i
O is a Dedekind Domain, every prime ideal is maximal. Thus the quotients κ(p ) := O /p
K i K i
and κ(p):=Z/pZ∼=F are fields, called the residue fields. Hence κ(p )|κ(p) is a field extension
p i
of finite degree.
Definition 1.11. The degree f := f (p |p) of the extension κ(p )|κ(p) is called the inertial
i i i i
degree of p over p.
i
The residue field κ(p) contains the residue field F , thus κ(p) is isomorphic to a finite field F
p pf
where f is the inertial degree of p over p.
Proposition 1.12. The ramification and inertial degrees satisfy the following equality
g
X
n= e f .
i i
i=1
Proof. See [Neu99, Proposition 8.2 in Chapter 1].
Let L be a number field containing K, such that [L:K] is finite. In a similar way as before, the
prime ideal p of OK has a decomposition pOL = Pe101···Peg0g in OL, where the Pi’s are prime
idealsofO . Inthesamewaywecanalsodefinetheramificationindexe(P |p)andtheinertial
L i
degree f(P |p).
i
Proposition 1.13. Let L|K|Q be a tower of number fields. Let p be a prime in Z, let p be
a prime ideal of O lying over p and P a prime ideal of O lying over p. Then we have
K L
e(P|p)=e(P|p)·e(p|p) and f(P|p)=f(P|p)·f(p|p).
Proof. See [Mol11, Theorem 5.1].
Finally, it is natural to ask about the number of primes that ramify in a finite extension of
number fields.
Theorem 1.14 (Minkowski). Let K|Q be a non trivial extension. Then there is at least one
ramified prime and the number of ramified primes is finite.
Proof. See [Neu99, Proposition 8.4 in Chapter 1] and [Neu99, Theorem 2.18 in Chapter 3].
Remark 1.15. For general number fields K|L it is still true that the number of prime ideals of
O that ramify in O is finite. However it is not necessarily true that there is a least one such
K L
prime.
1.3 Ramification of primes in Galois extensions
If K|Q is normal, the extension is Galois, since every number field is separable over Q. Let G
K
denote its Galois group, and p a prime ideal lying over p.
i
First we note that for any σ ∈ G we have σ(O ) = O . Indeed, if α ∈ O , it is the root
K K K K
of some polynomial xn+a xn−1+···+a x+a , where the a ∈ Z. Since Z is contained in
n−1 1 0 i
Q, the coefficients a are fixed by σ and σ(α) is also a root of the polynomial. Thus σ(α)∈O .
i K
Since σ ∈G fixes Q and σ(p )∩Z⊆Q, we have σ(p )∩Z=σ(p ∩Z)=p ∩Z=(p). Thus
K i i i i
σ(p ) is also a prime ideal lying over p. So G acts on the set of prime ideals p lying over p.
i K i
1 RAMIFICATION THEORY 8
Proposition 1.16. The group G acts transitively on the set of prime ideals lying over p.
K
Proof. See [Neu99, Proposition 9.1 in Chapter 1].
Proposition 1.17. Let K|Q be a finite Galois extension and p ,...,p be the prime ideals lying
1 g
over p. Then the ramification indices e = e(p |p) and inertial degrees f = f (p |p) are both
i i i i i
independant of i. Thus
e =···=e and f =···=f .
1 g 1 g
Proof. Letp beaprimelyingoverp. ByProposition1.16wehavep =σ (p )forsomeσ ∈G .
i i i 1 i K
Since σ fixes p⊆Q and O , we have
i K
pO =σ (pO )=σ (pe1pe2···peg)
K i K i 1 2 g
=σ (p )e1σ (p )e2···σ (p )eg
i 1 i 2 i g
=pe1σ (p )e2···σ (p )eg.
i i 2 i g
Thus for any i we have e =e . Moreover, the automorphism σ induces an isomorphism
i 1 i
κ(p )−→κ(p ), x mod p 7−→σ (x) mod σ (p ).
1 i 1 i i 1
This implies that f =f for any i.
i 1
From now on, let e:=e =···=e and f :=f =···=f . Thus the factorization of p becomes
1 g 1 g
(cid:0) (cid:1)e
pO = p ···p . Using Proposition 1.12 we deduce the following result.
K 1 g
Corollary 1.18. If K|Q is a Galois extension of degree n, then we have
n=efg.
(cid:0) (cid:1)e
Definition 1.19. If pO = p ···p is the factorization of p in O we say that
K 1 g K
1. p is ramified if e>1,
2. p is totally ramified if e=n,
3. p is unramified if e=1,
4. p splits completely if g =n.
1.4 Decomposition and Inertia groups
Let K|Q be a Galois extension of degree n with Galois group G and pO = (p ···p )e be
K K 1 g
the factorization of p in O . We saw in Proposition 1.16 that G acts transitively on the set
K K
of prime ideals {p ,...,p } lying over p. Let p := p be one of these ideals. The stabilizer of p
1 g i
under the action of G is called the decomposition group of p and is denoted by
K
D :={σ ∈G |σ(p)=p}.
p K
Its fixed field KDp is called the decomposition field of p.
Remark 1.20. The decomposition group fixes p as an ideal but not elementwise. This means
that for σ ∈D and x∈p we have σ(x)∈p but not necessarly σ(x)=x.
p
1 RAMIFICATION THEORY 9
Since every element σ ∈ D sends p to itself by definition, it induces an automorphism of the
p
residue fields
σ¯: κ(p)−→κ(p)
x mod p7−→σ(x) mod p.
Furthermore, the automorphism σ¯ fixes κ(p) since σ fixes Q, and hence σ¯ is an automorphism
of κ(p) fixing κ(p).
Proposition 1.21. The extension κ(p)|κ(p) is normal and the map
D −→Gal(κ(p)|κ(p))
p
σ 7−→σ¯
is a surjective homorphism.
Proof. See [Neu99, Proposition 9.4 in Chapter 1].
The kernel of this homomorphism is called the inertia group of p and denoted by I . Since the
p
homomorphism is surjective, we have the following isomorphism:
D /I ∼=Gal(κ(p)|κ(p)).
p p
The fixed field KIp of the inertia group is called the inertia field of p. An element σ ∈ Dp lies
in I if its image is the identity in Gal(κ(p)|κ(p)), hence the inertia group is exactly
p
I ={σ ∈D |∀x∈O σ(x)≡x mod p}.
p p K
WesawinSubsection1.2thattheresiduefieldsarefinitefieldsofcharacteristicp. Moreover,the
degreeoftheextensionκ(p)|κ(p)istheinertialdegreef =f(p|p). Thuswehaveanisomorphism
Gal(κ(p)|κ(p))∼=Gal(F |F ).
pf p
The Galois group of F |F is cyclic, generated by the automorphism σ : x 7→ xp, which is
pf p
called the Frobenius automorphism. A proof of this fact can for example be found in [DSD03,
Section 14.3].
Corollary 1.22. The quotient D /I is cyclic, generated by the coset of σ ∈D , such that
p p p
σ(x)≡xp mod p.
In other words, the image σ in D /I is the Frobenius automorphism.
p p
Proposition 1.23. We have
1. |D |=ef and (G :D )=g,
p K p
2. |I |=e and (D :I )=f.
p p p
Hence p is totally ramified in K if and only if I =G , and unramified if and only if I ={e}.
p K p
1 RAMIFICATION THEORY 10
Proof. 1. Since the action of G is transitive, the orbit is exactly the set of prime ideals lying
K
over p. Using the orbit stabilizer theorem and the fundamental equality from Corollary 1.18, we
obtain g·|D |=|G |=n=efg. This implies that |D |=ef and (G :D )= |GK| =g.
p K p K p |Dp|
2. From the definition of the inertial degree, we know that f = [κ(p) : κ(p)] = |Gal(κ(p)|κ(p))|.
Moreover, the quotient D /I is isomorphic to Gal(κ(p)|κ(p)). Thus |Gal(κ(p)|κ(p))| = |Dp|.
p p |Ip|
This implies that (D :I )=f. Since |D |=ef, we conclude that |I |=e.
p p p p
From the previous Proposition and the Fundamental Theorem of Galois theory, we deduce the
following Corollary.
Corollary 1.24. We have
1. [K :KIp]=e,
2. [KIp :KDp]=f,
3. [KDp :Q]=g.
Let F be a subfield of K containing Q and p :=p∩F be a prime of O lying under p and over
F F
p. Everything we did before for the extension K|Q can also be done for K|F. In this case, the
GaloisgroupGal(K|F)actstransitivelyonthesetofprimesofO lyingoverp ,andwedenote
K F
the decomposition group by D . Then, we get a surjective homormphism
p|p
F
D →Gal(κ(p)|κ(p )),
p|pF F
with kernel I =I ∩Gal(K|F). Moreover, we have |I |=e(p|p ).
p|pF p p|pF F
Proposition 1.25. Let K|Q be a finite Galois extension and p a prime of O . Let F be a
K
subfield of K and p :=p∩F. Then I is isomorphic to I /I .
F pF p p|pF
Proof. Letσ ∈I ,i.e. σ(x)−x∈pforeveryx∈O . RestrictingtoF,wehaveσ (cid:22) (x)−x∈p
p K F F
for every x ∈ O , since p = p∩F and O = O ∩F. Thus σ (cid:22) ∈ I . Consider the group
F F F K F pF
homomorphism
φ : I −→ I
σp 7−→ σp(cid:22)F .
F
The kernel of φ is I ∩Gal(K|F) = I , thus I /I ∼= Imφ ≤ I . Since the ramification
p p|p p p|p p
F F F
indices are multiplicative by Proposition 1.13, it follows that |I /I | = e(p|p) = e(p |p) =
|I |. Hence we have I ∼=I /I p p|pF e(p|pF) F
p p p p|p
F F F
From this Proposition we can deduce two important corollaries.
Corollary 1.26. The prime p is unramified in the inertia field KIp.
Proof. Let p⊂OK lying over p and pI :=p∩KIp. We know that Gal(K|KIp)=Ip from Galois
theory. Moreover, we have Ip|p = Ip∩Gal(K|KIp) = Ip. By Proposition 1.25, it follows that
I
Ip ={e} and thus p is unramified in KIp.
I
Corollary 1.27. Let K|Q be a finite Galois extension and F a subfield of K cointaining Q. If
p is totally ramified in K, then p is totally ramified in F.