Analogue of the Kronecker–Weber Theorem in positive characteristic

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Gabriel Villa Salvador

Introduction Cyclotomic function fields The maximal abelian extension of the rational function field The proof of David Hayes Witt vectors and the conductor The

Special Semester on Applications of Algebra and Number Theory Johann Radon Institute for Computational and Applied Mathematics (RICAM)

Linz, Austria, November 11 – 15, 2013

Analogue of the Kronecker–Weber Theorem in positive characteristic

Centro de Investagaci´on y de Estudios Avanzados del I.P.N., Departamento de Control Autom´atico,

E-mail:[email protected]

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Introduction Cyclotomic function fields The maximal abelian extension of the rational function field The proof of David Hayes Witt vectors and the conductor The Kronecker–

Weber–Hayes Theorem

Topic

1 Introduction

2 Cyclotomic function fields

3 The maximal abelian extension of the rational function field

4 The proof of David Hayes

5 Witt vectors and the conductor

6 The Kronecker–Weber–Hayes Theorem

7 Bibliography

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Introduction

We may understand byclass field theoryas the study of abelian extensions of global and local fields. In some sense, the simplest object of these two families of fields is the field of rational numbersQ. Therefore, one of the objectives in class field theory is to take care of the maximal abelian extension ofQ.

The first one to study the maximal abelian extension ofQas such was Leopold Kronecker in 1853 [1]. He claimed that every finite abelian extension ofQ was contained in a cyclotomic fieldQ(ζn) for somen∈N. The proof of Kronecker was not complete as he himself was aware.

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Introduction

We may understand byclass field theoryas the study of abelian extensions of global and local fields. In some sense, the simplest object of these two families of fields is the field of rational numbersQ. Therefore, one of the objectives in class field theory is to take care of the maximal abelian extension ofQ.

The first one to study the maximal abelian extension ofQ as such was Leopold Kronecker in 1853 [1]. He claimed that every finite abelian extension ofQwas contained in a cyclotomic fieldQ(ζn) for somen∈N. The proof of Kronecker was not complete as he himself was aware.

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Number fields

Henrich Weber provided a proof of Kronecker’s result in 1886 [3]. Weber’s proof was also incomplete but the gap was not noticed up to more than ninety years later by Olaf Neuman [3].

The result is now known as theKronecker–Weber Theorem.

David Hilbert gave a new proof of Kronecker’s original statement in 1896 [4]. This was the first correct complete proof of the theorem. However, as we mention above, Hilbert was not aware of Weber’s gap. Because of this some people call the result theKronecker–Weber–Hilbert Theorem. Hilbert’s Twelfth Problem is precisely to extend the Kronecker–Weber Theorem to any base number field.

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Number fields

Henrich Weber provided a proof of Kronecker’s result in 1886 [3]. Weber’s proof was also incomplete but the gap was not noticed up to more than ninety years later by Olaf Neuman [3].

The result is now known as theKronecker–Weber Theorem.

David Hilbert gave a new proof of Kronecker’s original statement in 1896 [4]. This was the first correct complete proof of the theorem. However, as we mention above, Hilbert was not aware of Weber’s gap. Because of this some people call the result theKronecker–Weber–Hilbert Theorem.

Hilbert’s Twelfth Problem is precisely to extend the Kronecker–Weber Theorem to any base number field.

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Congruence function fields

The analogue of the Kronecker–Weber Theorem for function fields is to find explicitly the maximal abelian extension of a rational function field with field of constants the finite field ofq elementsk=Fq(T).

One natural question here is if there exist something similar to cyclotomic fields in the case of function fields. Note that in full generality we have “cyclotomic” extensions of an arbitrary base fieldF, namely, F(ζn)where ζn denotes a generator of the groupW_n={ξ∈F¯ |ξⁿ= 1},F¯ denoting a fixed algebraic closure ofF. However, in our case, k(ζ_n)/k is just an extension of constants.

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Congruence function fields

The analogue of the Kronecker–Weber Theorem for function fields is to find explicitly the maximal abelian extension of a rational function field with field of constants the finite field ofq elementsk=Fq(T).

One natural question here is if there exist something similar to cyclotomic fields in the case of function fields. Note that in full generality we have “cyclotomic” extensions of an arbitrary base fieldF, namely, F(ζn)where ζn denotes a generator of the groupW_n={ξ∈F¯ |ξⁿ= 1},F¯ denoting a fixed algebraic closure ofF. However, in our case, k(ζ_n)/k is just an extension of constants.

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Congruence function fields 2

Leonard Carlitz established an analogue of cyclotomic number fields to the case of function fields. David Hayes [3] developed the ideas of Carlitz and he was able to describe explicitly the maximal abelian extensionA of k.

His result says that the maximal abelian extension of the rational function fieldFq(T) is the composite of three pairwise linearly disjoint extensions. Hayes’ description ofA is analogous to the Kronecker–Weber Theorem. Hayes’ approach to findA is the use of the

Artin–Takagi reciprocity law in class field theory.

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Congruence function fields 2

Leonard Carlitz established an analogue of cyclotomic number fields to the case of function fields. David Hayes [3] developed the ideas of Carlitz and he was able to describe explicitly the maximal abelian extensionA of k. His result says that the maximal abelian extension of the rational function fieldFq(T) is the composite of three pairwise linearly disjoint extensions.

Hayes’ description ofA is analogous to the Kronecker–Weber Theorem. Hayes’ approach to findA is the use of the

Artin–Takagi reciprocity law in class field theory.

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Congruence function fields 3

The main purpose of this talk is to present another approach to Hayes’ result. The main tools of this description is based on the Artin–Schreier–Witt theory ofp–cyclic extensions of fields of characteristicpand particularly the arithmetic of these extensions developed by Ernest Witt and Hermann Ludwig Schmid [2].

We may say that this approach is of combinatorial nature since, based on the results of Witt and Schmid, we compare the number of certain class of cyclic extensions with the number of such extensions contained inA. We find then that these two numbers are the same and from here the result follows.

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Congruence function fields 3

The main purpose of this talk is to present another approach to Hayes’ result. The main tools of this description is based on the Artin–Schreier–Witt theory ofp–cyclic extensions of fields of characteristicpand particularly the arithmetic of these extensions developed by Ernest Witt and Hermann Ludwig Schmid [2]. We may say that this approach is of combinatorial nature since, based on the results of Witt and Schmid, we compare the number of certain class of cyclic extensions with the number of such extensions contained inA. We find then that these two numbers are the same and from here the result follows.

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Cyclotomic function fields 1

We present the basic properties of the Carlitz–Hayes cyclotomic function fieds.

LetT be a transcendental fixed element over the finite field of q elements Fq and considerk:=Fq(T). Here the pole divisor p_∞ ofT ink is called the infinite prime. LetR_T :=Fq[T]be the ring of polynomials inT. Herek plays the role ofQand RT the role ofZ.

Since the fieldkconsists of two parts: Fq andT, we consider two special elements ofEnd_F_q(¯k): the Frobenius

automorphismϕof ¯k/Fq, andµT multiplication by T. More precisely, letϕ, µ_T ∈End_F_q(¯k) be given by

ϕ: ¯k→k¯ , µT: ¯k→¯k u7→u^q u7→T u.

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Cyclotomic function fields 1

ϕ: ¯k→k¯ , µT: ¯k→¯k u7→u^q u7→T u.

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Cyclotomic function fields 1

→ ¯ →¯

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Cyclotomic function fields 2

For anyM ∈R_T, the substitution T 7→ϕ+µ_T in M gives a ring homomorphismR_T −→^ξ End_F_q(¯k),

ξ(M(T)) =M(ϕ+µ_T). That is, ifu∈¯kandM ∈R_T, then

ξ(M)(u) =a_d(ϕ+µ_T)^d(u) +· · ·+a₁(ϕ+µ_T)(u) +a₀u

whereM(T) =a_dT^d+· · ·a1T+a0. In this way ¯kbecomes an R_T–module. The action is denoted as follows: ifM ∈R_T and u∈¯k,M◦u=ξ(M)(u) :=u^M.

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Cyclotomic function fields 2

For anyM ∈R_T, the substitution T 7→ϕ+µ_T in M gives a ring homomorphismR_T −→^ξ End_F_q(¯k),

ξ(M(T)) =M(ϕ+µ_T). That is, ifu∈¯kandM ∈R_T, then

ξ(M)(u) =a_d(ϕ+µ_T)^d(u) +· · ·+a₁(ϕ+µ_T)(u) +a₀u

whereM(T) =a_dT^d+· · ·a₁T+a₀. In this way ¯kbecomes an R_T–module. The action is denoted as follows: ifM ∈R_T and u∈¯k,M◦u=ξ(M)(u) :=u^M.

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Cyclotomic function fields 3

This action ofRT onk¯ is the analogue of the action of Zon Q¯^∗: n∈Z,x∈Q¯^∗,n◦x:=xⁿ. Of course the action ofRT is an additive action onk¯ andZacts multiplicatively onQ¯^∗.

The analogy of these two actions runs as follows. IfM ∈R_T, letΛM :={u∈¯k|u^M = 0}which is analogous to

Λ_m:={x∈Q¯^∗|x^m= 1},m∈Z. We have that Λ_M is an R_T–cyclic module. Indeed we haveΛ_M ∼=R_T/(M) as

RT–modules. A fixed generator of ΛM will be denoted byλM.

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Cyclotomic function fields 3

This action ofRT onk¯ is the analogue of the action of Zon Q¯^∗: n∈Z,x∈Q¯^∗,n◦x:=xⁿ. Of course the action ofRT is an additive action onk¯ andZacts multiplicatively onQ¯^∗. The analogy of these two actions runs as follows. IfM ∈RT, letΛM :={u∈¯k|u^M = 0}which is analogous to

Λ_m:={x∈Q¯^∗ |x^m= 1},m∈Z. We have that Λ_M is an RT–cyclic module. Indeed we haveΛM ∼=RT/(M) as

RT–modules. A fixed generator of ΛM will be denoted byλM.

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Cyclotomic function fields 4

Letk_M :=k(Λ_M) =k(λ_M). Thenk_M/k is an abelian

extension with Galois groupG_M := Gal(k_M/k)∼= R_T/(M)∗

the multiplicative group of invertible elements ofRT/(M).

Thus

[kM :k] =|G_M|=

RT/(M)∗

=: Φ(M). We have thatΦ(M) is a multiplicative function:

Φ(M N) = Φ(M)Φ(N) forM, N ∈RT with gcd(M, N) = 1. IfP ∈R_T is an irreducible polynomial and n∈N we have Φ(Pⁿ) =q^nd−q^(n−1)d=q^(n−1)d(q^d−1).

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Cyclotomic function fields 4

Thus

[kM :k] =|G_M|=

RT/(M)∗

=: Φ(M).

We have thatΦ(M) is a multiplicative function:

Φ(M N) = Φ(M)Φ(N) forM, N ∈RT with gcd(M, N) = 1. IfP ∈R_T is an irreducible polynomial and n∈N we have Φ(Pⁿ) =q^nd−q^(n−1)d=q^(n−1)d(q^d−1).

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Cyclotomic function fields 4

Thus

[kM :k] =|G_M|=

RT/(M)∗

=: Φ(M).

We have thatΦ(M) is a multiplicative function:

Φ(M N) = Φ(M)Φ(N) forM, N ∈RT with gcd(M, N) = 1.

IfP ∈R_T is an irreducible polynomial and n∈N we have Φ(Pⁿ) =q^nd−q^(n−1)d=q^(n−1)d(q^d−1).

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Cyclotomic function fields 5

The ramification in the extensionkM/kwhen M =Pⁿ is given by the following result.

Theorem

IfM =Pⁿ with P an irreducible polynomial inR_T, thenP is fully ramified inkPⁿ/k. We have

Φ(Pⁿ) =e_P = [k_Pⁿ :k] =q^(n−1)d(q^d−1), whered= degP. Any other finite prime inkis unramified in k_Pⁿ/k.

IfP =p∞,eP =e∞=ep∞ =q−1,fP =f∞=fp∞ = 1, h_P =h∞=hp∞ = Φ(M)/(q−1).

The extensionk_Pⁿ/k is a geometric extension, that is, the field of constants ofkPⁿ isFq and every subextensionk$K ⊆kPⁿ

is ramified.

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Cyclotomic function fields 5

The ramification in the extensionkM/kwhen M =Pⁿ is given by the following result.

Theorem

IfM =Pⁿ with P an irreducible polynomial inR_T, thenP is fully ramified inkPⁿ/k. We have

Φ(Pⁿ) =e_P = [k_Pⁿ :k] =q^(n−1)d(q^d−1), where d= degP.

Any other finite prime inkis unramified in k_Pⁿ/k.

IfP =p_∞,eP =e∞=ep∞ =q−1,fP =f∞=fp∞ = 1, h_P =h∞=h_p_∞ = Φ(M)/(q−1).

The extensionk_Pⁿ/k is a geometric extension, that is, the field of constants ofkPⁿ isFq and every subextension k$K ⊆kPⁿ

is ramified.

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Cyclotomic function fields 6

One important fact when we consider cyclotomic function fields, is the behavior ofp_∞ in any k_M/k where always e∞=q−1 andf∞= 1. In particularp∞ is alwaystamely ramified. Furthermore, for any subextensionL/K with

k⊆K ⊆L⊆k_M for some M ∈R_T, if the prime divisors ofK dividingp_∞ are unramified, then they are fully decomposed.

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The maximal abelian extension of k

LetA be the maximal abelian extension ofk. The expression ofA can be given explicitly, namely,A is explicitly generated for suitable finite extensions ofk, each one of which is generated by roots of an explicit polynomial. IndeedA is the composite of three pairwise linearly disjoint extensionsE/k, k_(T₎/kand k∞/k.

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First component

E/k: Consider the usual cyclotomic extensions ofk, that is, the constant extensions ofk. So E=S∞

n=1Fqⁿ(T). We have

GE := Gal(E/k)∼= ˆZ∼= Y

pprime

Zp,

whereZˆ is the Pr¨ufer ring andZp,pa prime number, is the ring ofp–adic numbers. We have that E/kis an unramified extension.

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Second component

k_(T₎/k: Now we consider all the Carlitz–Hayes cyclotomic function fields with respectp_∞,k_(T₎:=S

M∈R_Tk_M. We have G_T := Gal(k_(T₎/k)∼= lim_←

M∈R_T

R_T/(M)∗

.

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What is missing?

k∞/k: The fieldEk_(T₎ is an abelian extension ofkbut can not be the maximal one sincep_∞ is tamely ramified in Ek_(T₎/k and there exist abelian extensionsK/k wherep_∞ is wildly ramified. For instance, considerK =k(y) wherey^p−y=T. ThenK/k is a cyclic extension of degree p, wherep is the characteristic ofkandp_∞ is the only ramified prime in K/k and it is wildly ramified.

We change our “variable”T forT⁰ = 1/T and we now consider the cyclotomic function fields corresponding to the variableT⁰ instead ofT. Namely

k_(T⁰₎ =k_(1/T):= [

M⁰∈R_T0

k(Λ_M⁰), R_T⁰ =Fq[T⁰].

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What is missing?

k∞/k: The fieldEk_(T₎ is an abelian extension ofkbut can not be the maximal one sincep_∞ is tamely ramified in Ek_(T₎/k and there exist abelian extensionsK/k wherep_∞ is wildly ramified. For instance, considerK =k(y) wherey^p−y=T. ThenK/k is a cyclic extension of degree p, wherep is the characteristic ofkandp_∞ is the only ramified prime in K/k and it is wildly ramified.

We change our “variable”T forT⁰ = 1/T and we now consider the cyclotomic function fields corresponding to the variableT⁰ instead ofT. Namely

k_(T⁰₎=k_(1/T):= [

M⁰∈R_T0

k(Λ_M⁰), R_T⁰ =Fq[T⁰].

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k

_(T₎

and k

_(T⁰₎

are not linearly disjoint

We have thatk_(T⁰₎ shares much withk_(T₎. For instance, if q=p²,p >3andz^p−z= _(T^T_+1)(T²^+T⁺¹₊₂₎, then

K:=k(z)⊆k_(T₎∩k_(T⁰₎.

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Third component

In order to find some subextension ofk_(T⁰₎ linearly disjoint to k_(T₎, consider L_T⁰ :=S∞

m=1k(Λ_(T⁰₎^m). InL_T⁰/kthe only ramified primes arep_∞, which is totally ramified, and the prime p₀ corresponding to the cero divisor of T. The primep₀ is now the infinite prime ink_(T⁰₎ and it is tamely ramified with

ramification indexq−1.

LetG⁰₀ =F^∗_q = RT⁰/(T⁰)∗

be the inertia group ofp₀. Then k∞:=L^G

0 0

T⁰ is an abelian extension of kwherep_∞ is the only ramified prime and it is totally wildly ramified, that is, for any finite extensionF/k,k$F ⊆k∞, thenp_∞ is totally ramified in F and has no tame ramification. This is equivalent to have that the Galois group and the first ramification group are the same.

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Third component

In order to find some subextension ofk_(T⁰₎ linearly disjoint to k_(T₎, consider L_T⁰ :=S∞

m=1k(Λ_(T⁰₎^m). InL_T⁰/kthe only ramified primes arep_∞, which is totally ramified, and the prime p₀ corresponding to the cero divisor of T. The primep₀ is now the infinite prime ink_(T⁰₎ and it is tamely ramified with

ramification indexq−1. LetG⁰₀ =F^∗_q = RT⁰/(T⁰)∗

be the inertia group ofp₀. Then k∞:=L^G

0 0

T⁰ is an abelian extension of kwherep_∞ is the only ramified prime and it is totally wildly ramified, that is, for any finite extensionF/k,k$F ⊆k∞, thenp_∞ is totally ramified in F and has no tame ramification.

This is equivalent to have that the Galois group and the first

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Why is the maximal abelian extension?

The extensionB:=k_(T₎·k∞·E is an abelian extension with k_(T₎, k∞, E pairwise linearly disjoint. WhyA=B? Hayes’

proof answers this question.

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Decomposition of the idele group

LetA=k_(T₎k∞E. The question is why A is the maximal abelian extension ofk. First, Hayes constructed a group homomorphismψ:J_k →Gal(A/k), whereJ_k es the idele group ofk. Since k_(T₎, k∞ andE are pairwise linearly disjoint, we haveGal(A/k)∼=G_(T₎×G∞×GE where

G_(T₎ = Gal(k_(T₎/k),G∞= Gal(k∞/k) and G_E = Gal(E/k)∼= ˆZ.

For his construction, Hayes decomposedJ =Jk as the direct product of four subgroups and definedψ directly in each one of the four subgroups. Indeed, the map is trivial on one factor and the other three factors map intoG_(T₎,G∞ andGE

respectively. The factorization was of the following type: J ∼=k^∗×U_T ×k_p⁽¹⁾_∞×Z

both algebraically and topologically.

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Decomposition of the idele group

respectively. The factorization was of the following type:

J ∼=k^∗×U_T ×k_p⁽¹⁾_∞×Z

both algebraically and topologically.

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Decomposition of the idele group

respectively. The factorization was of the following type:

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Isomorphisms

The next step in Hayes’ construction consisted in proving that there exist natural isomorphismsψT:UT →G_(T₎ and

ψ∞:k⁽¹⁾_p_∞ →G∞∼={f(1/T)∈Fq[[1/T]]|f(0) = 1}, both algebraically and topologically. Nowψ_Z:Z→GE ∼= ˆZis the map such thatψ_Z(1) is the Frobenius automorphism.

Thereforeψ_Z is a dense continuous monomorphism.

In short, we have

ψ_T:U_T −→^∼⁼ G_(T₎, ψ∞:k_p⁽¹⁾_∞ −→^∼⁼ G∞ and ψ_Z:Z,→Zˆ.

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Isomorphisms

In short, we have

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Isomorphisms

In short, we have

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End of the Hayes’ proof

The final step in Hayes’ proof was to show that with these isomorphisms, the Reciprocity Law of Artin–Takagi gives that Ais the maximal abelian extension of k.

Hayes also proved thatA=k_(T₎k_(T⁰₎ with T⁰ = 1/T. However, as we have noticed,k_(T₎ and k_(T⁰₎ are not linearly disjoint.

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End of the Hayes’ proof

The final step in Hayes’ proof was to show that with these isomorphisms, the Reciprocity Law of Artin–Takagi gives that Ais the maximal abelian extension of k.

Hayes also proved thatA=k_(T₎k_(T⁰₎ with T⁰ = 1/T. However, as we have noticed,k_(T₎ and k_(T⁰₎ are not linearly disjoint.

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The conductor

LetK =k(~y) be such that℘~y=~y^p −^• ~y=β~∈W_n(k), βi

= _p^c_λiⁱ with λi ≥0 and ifλi >0, then gcd(ci,p) = 1 and gcd(λi, p) = 1where pis the prime divisor associated to P. LetM_n:= max

1≤i≤n{pⁿ⁻ⁱλ_i}. Note that M_i = max{pM_i−1, λ_i}, M₁ < M₂ <· · ·< M_n. Then

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The conductor according to Schmid

Theorem (Schmid [2])

With the above conditions we have that the conductor ofK/k is

f_K =P^Mⁿ⁺¹.

Corollary

LetK/k be a cyclic extension of degreepⁿwith K ⊆k(λ_P^α) for someα∈N. ThenMn+ 1≤α.

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The conductor according to Schmid

Theorem (Schmid [2])

With the above conditions we have that the conductor ofK/k is

f_K =P^Mⁿ⁺¹.

Corollary

LetK/k be a cyclic extension of degreepⁿwith K ⊆k(λ_P^α) for someα ∈N. ThenMn+ 1≤α.

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The Kronecker–Weber–Hayes Theorem

To prove the Kronecker–Weber–Hayes Theorem it suffices to prove that any finite abelian extension ofk is contained in k_NFq^mk_n for someN ∈R_T,m, n∈Nand where

k_n:= Sn+1

r=1k(λ_T^−r)G⁰₀

=k(λ_T⁻ⁿ⁻¹)^G⁰⁰.

It suffices to prove this when the abelian extension is cyclic of order either relatively prime top or of orderp^u for someu∈N. The Kronecker–Weber Theorem will be a consequence of the following facts.

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The Kronecker–Weber–Hayes Theorem

k_n:= Sn+1

=k(λ_T⁻ⁿ⁻¹)^G⁰⁰.

It suffices to prove this when the abelian extension is cyclic of order either relatively prime top or of orderp^u for some u∈N.

The Kronecker–Weber Theorem will be a consequence of the following facts.

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The Kronecker–Weber–Hayes Theorem

k_n:= Sn+1

=k(λ_T⁻ⁿ⁻¹)^G⁰⁰.

It suffices to prove this when the abelian extension is cyclic of order either relatively prime top or of orderp^u for some u∈N. The Kronecker–Weber Theorem will be a consequence of the following facts.

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Reduction steps

2 (a) IfK/k is a finite tamely ramified abelian extension whereP₁, . . . , P_r∈R⁺_T and possibly p_∞ are the ramified primes, then

K ⊆Fq^mk(ΛP1···P_r) for some m∈N.

2 (b) If K/k is a cyclic extension of degreepⁿ where P ∈R_T⁺ is the only ramified prime,P is totally ramified and p_∞ is fully decomposed, thenK ⊆k(Λ_P^α) for some α∈N.

2 (c) If K/k is a cyclic extension of degreepⁿ where

P ∈R_T⁺is the only ramified prime, then K⊆Fq^pmk(ΛP^α) for somem, α∈N.

2 (d) Similarly for p_∞, that is, if K/kis a cyclic extension of degree pⁿ andp∞ is the only ramified prime, then

K ⊆Fq^pmkα for somem, α∈N.

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Reduction steps

2 (b) If K/k is a cyclic extension of degreepⁿ where P ∈R_T⁺ is the only ramified prime,P is totally ramified andp_∞ is fully decomposed, thenK ⊆k(Λ_P^α) for some α∈N.

P ∈R_T⁺is the only ramified prime, then K⊆Fq^pmk(ΛP^α) for somem, α∈N.

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Reduction steps

2 (b) If K/k is a cyclic extension of degreepⁿ where P ∈R_T⁺ is the only ramified prime,P is totally ramified andp_∞ is fully decomposed, thenK ⊆k(Λ_P^α) for some α∈N.

P ∈R_T⁺is the only ramified prime, then K⊆Fq^pmk(Λ_P^α) for somem, α∈N.