Cyclotomic function fields 1 We present the basic properties of the Carlitz–Hayes cyclotomic Gabriel function fieds. Villa Salvador Let T be a transcendental fixed element over the finite field of Introduction q elements F q and consider k := F q ( T ) . Here the pole divisor Cyclotomic p ∞ of T in k is called the infinite prime . Let R T := F q [ T ] be function fields the ring of polynomials in T . Here k plays the role of Q and The maximal abelian R T the role of Z . extension of the rational Since the field k consists of two parts: F q and T , we consider function field two special elements of End F q (¯ k ) : the Frobenius The proof of automorphism ϕ of ¯ David Hayes k/ F q , and µ T multiplication by T . More Witt vectors precisely, let ϕ, µ T ∈ End F q (¯ k ) be given by and the conductor ϕ : ¯ k → ¯ µ T : ¯ k → ¯ The k , k Kronecker– Weber–Hayes u �→ u q u �→ Tu. Theorem Bibliography 8 / 53
Cyclotomic function fields 2 Gabriel Villa Salvador For any M ∈ R T , the substitution T �→ ϕ + µ T in M gives a Introduction ξ → End F q (¯ ring homomorphism R T − k ) , Cyclotomic function fields ξ ( M ( T )) = M ( ϕ + µ T ) . That is, if u ∈ ¯ k and M ∈ R T , then The maximal abelian extension of the rational function field The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 9 / 53
Cyclotomic function fields 2 Gabriel Villa Salvador For any M ∈ R T , the substitution T �→ ϕ + µ T in M gives a Introduction ξ → End F q (¯ ring homomorphism R T − k ) , Cyclotomic function fields ξ ( M ( T )) = M ( ϕ + µ T ) . That is, if u ∈ ¯ k and M ∈ R T , then The maximal abelian ξ ( M )( u ) = a d ( ϕ + µ T ) d ( u ) + · · · + a 1 ( ϕ + µ T )( u ) + a 0 u extension of the rational function field where M ( T ) = a d T d + · · · a 1 T + a 0 . In this way ¯ k becomes an The proof of David Hayes R T –module. The action is denoted as follows: if M ∈ R T and Witt vectors u ∈ ¯ k , M ◦ u = ξ ( M )( u ) := u M . and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 9 / 53
Cyclotomic function fields 3 Gabriel Villa Salvador This action of R T on ¯ k is the analogue of the action of Z on Introduction Q ∗ : n ∈ Z , x ∈ ¯ ¯ Q ∗ , n ◦ x := x n . Of course the action of R T is Cyclotomic function fields an additive action on ¯ k and Z acts multiplicatively on ¯ Q ∗ . The maximal abelian extension of the rational function field The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 10 / 53
Cyclotomic function fields 3 Gabriel Villa Salvador This action of R T on ¯ k is the analogue of the action of Z on Introduction Q ∗ : n ∈ Z , x ∈ ¯ ¯ Q ∗ , n ◦ x := x n . Of course the action of R T is Cyclotomic function fields an additive action on ¯ k and Z acts multiplicatively on ¯ Q ∗ . The maximal The analogy of these two actions runs as follows. If M ∈ R T , abelian extension of k | u M = 0 } which is analogous to let Λ M := { u ∈ ¯ the rational function field Q ∗ | x m = 1 } , m ∈ Z . We have that Λ M is an Λ m := { x ∈ ¯ The proof of R T –cyclic module. Indeed we have Λ M ∼ = R T / ( M ) as David Hayes Witt vectors R T –modules. A fixed generator of Λ M will be denoted by λ M . and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 10 / 53
Cyclotomic function fields 4 Gabriel Villa Salvador Let k M := k (Λ M ) = k ( λ M ) . Then k M /k is an abelian extension with Galois group G M := Gal( k M /k ) ∼ � ∗ � R T / ( M ) = Introduction the multiplicative group of invertible elements of R T / ( M ) . 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 Bibliography 11 / 53
Cyclotomic function fields 4 Gabriel Villa Salvador Let k M := k (Λ M ) = k ( λ M ) . Then k M /k is an abelian extension with Galois group G M := Gal( k M /k ) ∼ � ∗ � R T / ( M ) = Introduction the multiplicative group of invertible elements of R T / ( M ) . Cyclotomic function fields Thus The maximal abelian � ∗ � � =: Φ( M ) . extension of � �� [ k M : k ] = | G M | = R T / ( M ) the rational function field The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 11 / 53
Cyclotomic function fields 4 Gabriel Villa Salvador Let k M := k (Λ M ) = k ( λ M ) . Then k M /k is an abelian extension with Galois group G M := Gal( k M /k ) ∼ � ∗ � R T / ( M ) = Introduction the multiplicative group of invertible elements of R T / ( M ) . Cyclotomic function fields Thus The maximal abelian � ∗ � � =: Φ( M ) . extension of � �� [ k M : k ] = | G M | = R T / ( M ) the rational function field The proof of We have that Φ( M ) is a multiplicative function: David Hayes Φ( MN ) = Φ( M )Φ( N ) for M, N ∈ R T with gcd( M, N ) = 1 . Witt vectors and the If P ∈ R T is an irreducible polynomial and n ∈ N we have conductor Φ( P n ) = q nd − q ( n − 1) d = q ( n − 1) d ( q d − 1) . The Kronecker– Weber–Hayes Theorem Bibliography 11 / 53
Cyclotomic function fields 5 The ramification in the extension k M /k when M = P n is given Gabriel Villa by the following result. 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 Kronecker– Weber–Hayes Theorem Bibliography 12 / 53
Cyclotomic function fields 5 The ramification in the extension k M /k when M = P n is given Gabriel Villa by the following result. Salvador Introduction Theorem Cyclotomic function fields If M = P n with P an irreducible polynomial in R T , then P is The maximal fully ramified in k P n /k . We have abelian extension of Φ( P n ) = e P = [ k P n : k ] = q ( n − 1) d ( q d − 1) , where d = deg P . the rational function field Any other finite prime in k is unramified in k P n /k . The proof of If P = p ∞ , e P = e ∞ = e p ∞ = q − 1 , f P = f ∞ = f p ∞ = 1 , David Hayes h P = h ∞ = h p ∞ = Φ( M ) / ( q − 1) . Witt vectors and the The extension k P n /k is a geometric extension, that is, the field conductor of constants of k P n is F q and every subextension k � K ⊆ k P n The Kronecker– is ramified. Weber–Hayes Theorem Bibliography 12 / 53
Cyclotomic function fields 6 Gabriel Villa Salvador Introduction One important fact when we consider cyclotomic function Cyclotomic function fields fields, is the behavior of p ∞ in any k M /k where always The maximal e ∞ = q − 1 and f ∞ = 1 . In particular p ∞ is always tamely abelian extension of ramified. Furthermore, for any subextension L/K with the rational function field k ⊆ K ⊆ L ⊆ k M for some M ∈ R T , if the prime divisors of K The proof of dividing p ∞ are unramified, then they are fully decomposed. David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 13 / 53
The maximal abelian extension of k Gabriel Villa Salvador Introduction Let A be the maximal abelian extension of k . The expression Cyclotomic function fields of A can be given explicitly, namely, A is explicitly generated The maximal for suitable finite extensions of k , each one of which is abelian extension of generated by roots of an explicit polynomial. Indeed A is the the rational function field composite of three pairwise linearly disjoint extensions E/k , The proof of k ( T ) /k and k ∞ /k . David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 14 / 53
First component Gabriel Villa Salvador E/k : Consider the usual cyclotomic extensions of k , that is, Introduction the constant extensions of k . So E = � ∞ n =1 F q n ( T ) . We have Cyclotomic function fields G E := Gal( E/k ) ∼ = ˆ Z ∼ � The maximal Z p , = abelian extension of p prime the rational function field where ˆ Z is the Pr¨ ufer ring and Z p , p a prime number, is the The proof of David Hayes ring of p –adic numbers. We have that E/k is an unramified Witt vectors and the extension. conductor The Kronecker– Weber–Hayes Theorem Bibliography 15 / 53
Second component Gabriel Villa Salvador Introduction k ( T ) /k : Now we consider all the Carlitz–Hayes cyclotomic Cyclotomic function fields function fields with respect p ∞ , k ( T ) := � M ∈ R T k M . We have The maximal abelian extension of G T := Gal( k ( T ) /k ) ∼ � ∗ . � lim R T / ( M ) = the rational ← function field M ∈ R T The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 16 / 53
What is missing? k ∞ /k : The field Ek ( T ) is an abelian extension of k but can not Gabriel Villa be the maximal one since p ∞ is tamely ramified in Ek ( T ) /k Salvador and there exist abelian extensions K/k where p ∞ is wildly Introduction ramified. For instance, consider K = k ( y ) where y p − y = T . Cyclotomic function fields Then K/k is a cyclic extension of degree p , where p is the The maximal characteristic of k and p ∞ is the only ramified prime in K/k abelian extension of and it is wildly ramified. the rational function field The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 17 / 53
What is missing? k ∞ /k : The field Ek ( T ) is an abelian extension of k but can not Gabriel Villa be the maximal one since p ∞ is tamely ramified in Ek ( T ) /k Salvador and there exist abelian extensions K/k where p ∞ is wildly Introduction ramified. For instance, consider K = k ( y ) where y p − y = T . Cyclotomic function fields Then K/k is a cyclic extension of degree p , where p is the The maximal characteristic of k and p ∞ is the only ramified prime in K/k abelian extension of and it is wildly ramified. the rational We change our “variable” T for T ′ = 1 /T and we now consider function field The proof of the cyclotomic function fields corresponding to the variable T ′ David Hayes instead of T . Namely Witt vectors and the conductor � R T ′ = F q [ T ′ ] . k ( T ′ ) = k (1 /T ) := k (Λ M ′ ) , The Kronecker– M ′ ∈ R T ′ Weber–Hayes Theorem Bibliography 17 / 53
k ( T ) and k ( T ′ ) are not linearly disjoint Gabriel Villa Salvador Introduction Cyclotomic function fields We have that k ( T ′ ) shares much with k ( T ) . For instance, if The maximal q = p 2 , p > 3 and z p − z = T 2 + T +1 ( T +1)( T +2) , then abelian extension of the rational K := k ( z ) ⊆ k ( T ) ∩ k ( T ′ ) . function field The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 18 / 53
Third component Gabriel In order to find some subextension of k ( T ′ ) linearly disjoint to Villa Salvador k ( T ) , consider L T ′ := � ∞ m =1 k (Λ ( T ′ ) m ) . In L T ′ /k the only Introduction ramified primes are p ∞ , which is totally ramified, and the prime Cyclotomic p 0 corresponding to the cero divisor of T . The prime p 0 is now function fields the infinite prime in k ( T ′ ) and it is tamely ramified with The maximal abelian ramification index q − 1 . extension of the rational function field The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 19 / 53
Third component Gabriel In order to find some subextension of k ( T ′ ) linearly disjoint to Villa Salvador k ( T ) , consider L T ′ := � ∞ m =1 k (Λ ( T ′ ) m ) . In L T ′ /k the only Introduction ramified primes are p ∞ , which is totally ramified, and the prime Cyclotomic p 0 corresponding to the cero divisor of T . The prime p 0 is now function fields the infinite prime in k ( T ′ ) and it is tamely ramified with The maximal � ∗ be the abelian ramification index q − 1 . Let G ′ 0 = F ∗ R T ′ / ( T ′ ) � q = extension of the rational inertia group of p 0 . Then k ∞ := L G ′ function field T ′ is an abelian extension of 0 The proof of k where p ∞ is the only ramified prime and it is totally wildly David Hayes ramified, that is, for any finite extension F/k , k � F ⊆ k ∞ , Witt vectors and the then p ∞ is totally ramified in F and has no tame ramification. conductor This is equivalent to have that the Galois group and the first The Kronecker– ramification group are the same. Weber–Hayes Theorem Bibliography 19 / 53
Why is the maximal abelian extension? Gabriel Villa Salvador Introduction Cyclotomic function fields The extension B := k ( T ) · k ∞ · E is an abelian extension with The maximal k ( T ) , k ∞ , E pairwise linearly disjoint. Why A = B ? Hayes’ abelian extension of proof answers this question. the rational function field The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 20 / 53
Decomposition of the idele group Let A = k ( T ) k ∞ E . The question is why A is the maximal Gabriel abelian extension of k . First, Hayes constructed a group Villa Salvador homomorphism ψ : J k → Gal( A/k ) , where J k es the idele group of k . Since k ( T ) , k ∞ and E are pairwise linearly disjoint, Introduction we have Gal( A/k ) ∼ = G ( T ) × G ∞ × G E where Cyclotomic function fields G ( T ) = Gal( k ( T ) /k ) , G ∞ = Gal( k ∞ /k ) and The maximal G E = Gal( E/k ) ∼ = ˆ abelian Z . extension of the rational function field The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 21 / 53
Decomposition of the idele group Let A = k ( T ) k ∞ E . The question is why A is the maximal Gabriel abelian extension of k . First, Hayes constructed a group Villa Salvador homomorphism ψ : J k → Gal( A/k ) , where J k es the idele group of k . Since k ( T ) , k ∞ and E are pairwise linearly disjoint, Introduction we have Gal( A/k ) ∼ = G ( T ) × G ∞ × G E where Cyclotomic function fields G ( T ) = Gal( k ( T ) /k ) , G ∞ = Gal( k ∞ /k ) and The maximal G E = Gal( E/k ) ∼ = ˆ abelian Z . extension of the rational For his construction, Hayes decomposed J = J k as the direct function field product of four subgroups and defined ψ directly in each one of The proof of David Hayes the four subgroups. Indeed, the map is trivial on one factor and Witt vectors the other three factors map into G ( T ) , G ∞ and G E and the conductor respectively. The factorization was of the following type: The Kronecker– Weber–Hayes Theorem Bibliography 21 / 53
Decomposition of the idele group Let A = k ( T ) k ∞ E . The question is why A is the maximal Gabriel abelian extension of k . First, Hayes constructed a group Villa Salvador homomorphism ψ : J k → Gal( A/k ) , where J k es the idele group of k . Since k ( T ) , k ∞ and E are pairwise linearly disjoint, Introduction we have Gal( A/k ) ∼ = G ( T ) × G ∞ × G E where Cyclotomic function fields G ( T ) = Gal( k ( T ) /k ) , G ∞ = Gal( k ∞ /k ) and The maximal G E = Gal( E/k ) ∼ = ˆ abelian Z . extension of the rational For his construction, Hayes decomposed J = J k as the direct function field product of four subgroups and defined ψ directly in each one of The proof of David Hayes the four subgroups. Indeed, the map is trivial on one factor and Witt vectors the other three factors map into G ( T ) , G ∞ and G E and the conductor respectively. The factorization was of the following type: The Kronecker– = k ∗ × U T × k (1) J ∼ p ∞ × Z Weber–Hayes Theorem Bibliography both algebraically and topologically. 21 / 53
Isomorphisms Gabriel Villa Salvador The next step in Hayes’ construction consisted in proving that there exist natural isomorphisms ψ T : U T → G ( T ) and Introduction Cyclotomic ψ ∞ : k (1) p ∞ → G ∞ ∼ = { f (1 /T ) ∈ F q [[1 /T ]] | f (0) = 1 } , both function fields algebraically and topologically. Now ψ Z : Z → G E ∼ = ˆ Z is the The maximal abelian map such that ψ Z (1) is the Frobenius automorphism. extension of the rational Therefore ψ Z is a dense continuous monomorphism. function field The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 22 / 53
Isomorphisms Gabriel Villa Salvador The next step in Hayes’ construction consisted in proving that there exist natural isomorphisms ψ T : U T → G ( T ) and Introduction Cyclotomic ψ ∞ : k (1) p ∞ → G ∞ ∼ = { f (1 /T ) ∈ F q [[1 /T ]] | f (0) = 1 } , both function fields algebraically and topologically. Now ψ Z : Z → G E ∼ = ˆ Z is the The maximal abelian map such that ψ Z (1) is the Frobenius automorphism. extension of the rational Therefore ψ Z is a dense continuous monomorphism. function field In short, we have The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 22 / 53
Isomorphisms Gabriel Villa Salvador The next step in Hayes’ construction consisted in proving that there exist natural isomorphisms ψ T : U T → G ( T ) and Introduction Cyclotomic ψ ∞ : k (1) p ∞ → G ∞ ∼ = { f (1 /T ) ∈ F q [[1 /T ]] | f (0) = 1 } , both function fields algebraically and topologically. Now ψ Z : Z → G E ∼ = ˆ Z is the The maximal abelian map such that ψ Z (1) is the Frobenius automorphism. extension of the rational Therefore ψ Z is a dense continuous monomorphism. function field In short, we have The proof of David Hayes Witt vectors ∼ ∼ = ψ ∞ : k (1) = → ˆ ψ T : U T − → G ( T ) , − → G ∞ and ψ Z : Z ֒ Z . and the p ∞ conductor The Kronecker– Weber–Hayes Theorem Bibliography 22 / 53
End of the Hayes’ proof Gabriel Villa Salvador Introduction Cyclotomic The final step in Hayes’ proof was to show that with these function fields isomorphisms, the Reciprocity Law of Artin–Takagi gives that The maximal abelian A is the maximal abelian extension of k . extension of the rational function field The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 23 / 53
End of the Hayes’ proof Gabriel Villa Salvador Introduction Cyclotomic The final step in Hayes’ proof was to show that with these function fields isomorphisms, the Reciprocity Law of Artin–Takagi gives that The maximal abelian A is the maximal abelian extension of k . extension of Hayes also proved that A = k ( T ) k ( T ′ ) with T ′ = 1 /T . However, the rational function field as we have noticed, k ( T ) and k ( T ′ ) are not linearly disjoint. The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 23 / 53
The conductor Gabriel Villa Salvador Introduction • y = � y p Let K = k ( � y ) be such that ℘� y = � − � β ∈ W n ( k ) , Cyclotomic function fields c i � � β i = p λi with λ i ≥ 0 and if λ i > 0 , then gcd( c i , p ) = 1 and The maximal abelian gcd( λ i , p ) = 1 where p is the prime divisor associated to P . extension of 1 ≤ i ≤ n { p n − i λ i } . Note that M i = max { pM i − 1 , λ i } , the rational Let M n := max function field M 1 < M 2 < · · · < M n . Then The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 24 / 53
The conductor according to Schmid Gabriel Villa Salvador Theorem (Schmid [2]) Introduction Cyclotomic With the above conditions we have that the conductor of K/k function fields is The maximal f K = P M n +1 . abelian extension of the rational function field The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 25 / 53
The conductor according to Schmid Gabriel Villa Salvador Theorem (Schmid [2]) Introduction Cyclotomic With the above conditions we have that the conductor of K/k function fields is The maximal f K = P M n +1 . abelian extension of the rational function field Corollary The proof of David Hayes Let K/k be a cyclic extension of degree p n with K ⊆ k ( λ P α ) Witt vectors and the for some α ∈ N . Then M n + 1 ≤ α . conductor The Kronecker– Weber–Hayes Theorem Bibliography 25 / 53
The Kronecker–Weber–Hayes Theorem Gabriel Villa Salvador To prove the Kronecker–Weber–Hayes Theorem it suffices to Introduction prove that any finite abelian extension of k is contained in Cyclotomic function fields k N F q m k n for some N ∈ R T , m, n ∈ N and where The maximal � G ′ 0 = k ( λ T − n − 1 ) G ′ � � n +1 abelian k n := r =1 k ( λ T − r ) 0 . extension of the rational function field The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 26 / 53
The Kronecker–Weber–Hayes Theorem Gabriel Villa Salvador To prove the Kronecker–Weber–Hayes Theorem it suffices to Introduction prove that any finite abelian extension of k is contained in Cyclotomic function fields k N F q m k n for some N ∈ R T , m, n ∈ N and where The maximal � G ′ 0 = k ( λ T − n − 1 ) G ′ � � n +1 abelian k n := r =1 k ( λ T − r ) 0 . extension of the rational It suffices to prove this when the abelian extension is cyclic of function field order either relatively prime to p or of order p u for some u ∈ N . The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 26 / 53
The Kronecker–Weber–Hayes Theorem Gabriel Villa Salvador To prove the Kronecker–Weber–Hayes Theorem it suffices to Introduction prove that any finite abelian extension of k is contained in Cyclotomic function fields k N F q m k n for some N ∈ R T , m, n ∈ N and where The maximal � G ′ 0 = k ( λ T − n − 1 ) G ′ � � n +1 abelian k n := r =1 k ( λ T − r ) 0 . extension of the rational It suffices to prove this when the abelian extension is cyclic of function field order either relatively prime to p or of order p u for some u ∈ N . The proof of David Hayes The Kronecker–Weber Theorem will be a consequence of the Witt vectors following facts. and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 26 / 53
Reduction steps ✒ (a) If K/k is a finite tamely ramified abelian extension Gabriel where P 1 , . . . , P r ∈ R + T and possibly p ∞ are the ramified Villa Salvador primes, then Introduction K ⊆ F q m k (Λ P 1 ··· P r ) for some m ∈ N . 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 Bibliography 27 / 53
Reduction steps ✒ (a) If K/k is a finite tamely ramified abelian extension Gabriel where P 1 , . . . , P r ∈ R + T and possibly p ∞ are the ramified Villa Salvador primes, then Introduction K ⊆ F q m k (Λ P 1 ··· P r ) for some m ∈ N . Cyclotomic function fields ✒ (b) If K/k is a cyclic extension of degree p n where The maximal P ∈ R + abelian T is the only ramified prime, P is totally ramified extension of and p ∞ is fully decomposed, then K ⊆ k (Λ P α ) for some the rational function field α ∈ N . The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 27 / 53
Reduction steps ✒ (a) If K/k is a finite tamely ramified abelian extension Gabriel where P 1 , . . . , P r ∈ R + T and possibly p ∞ are the ramified Villa Salvador primes, then Introduction K ⊆ F q m k (Λ P 1 ··· P r ) for some m ∈ N . Cyclotomic function fields ✒ (b) If K/k is a cyclic extension of degree p n where The maximal P ∈ R + abelian T is the only ramified prime, P is totally ramified extension of and p ∞ is fully decomposed, then K ⊆ k (Λ P α ) for some the rational function field α ∈ N . The proof of ✒ (c) If K/k is a cyclic extension of degree p n where David Hayes P ∈ R + Witt vectors T is the only ramified prime, then K ⊆ F q pm k (Λ P α ) and the conductor for some m, α ∈ N . The Kronecker– Weber–Hayes Theorem Bibliography 27 / 53
Reduction steps ✒ (a) If K/k is a finite tamely ramified abelian extension Gabriel where P 1 , . . . , P r ∈ R + T and possibly p ∞ are the ramified Villa Salvador primes, then Introduction K ⊆ F q m k (Λ P 1 ··· P r ) for some m ∈ N . Cyclotomic function fields ✒ (b) If K/k is a cyclic extension of degree p n where The maximal P ∈ R + abelian T is the only ramified prime, P is totally ramified extension of and p ∞ is fully decomposed, then K ⊆ k (Λ P α ) for some the rational function field α ∈ N . The proof of ✒ (c) If K/k is a cyclic extension of degree p n where David Hayes P ∈ R + Witt vectors T is the only ramified prime, then K ⊆ F q pm k (Λ P α ) and the conductor for some m, α ∈ N . The ✒ (d) Similarly for p ∞ , that is, if K/k is a cyclic extension of Kronecker– degree p n and p ∞ is the only ramified prime, then Weber–Hayes Theorem K ⊆ F q pm k α for some m, α ∈ N . Bibliography 27 / 53
Tame ramification Gabriel Villa Salvador For the part (a), first we observe 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 Bibliography 28 / 53
Tame ramification Gabriel Villa Salvador For the part (a), first we observe Introduction Cyclotomic function fields Proposici´ on The maximal abelian Let P ∈ R + T tamely ramified in K/k . If e is the ramification extension of index of P in K , we have e | q d − 1 where d = deg P . the rational function field The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 28 / 53
Tame ramification Gabriel Villa Salvador For the part (a), first we observe Introduction Cyclotomic function fields Proposici´ on The maximal abelian Let P ∈ R + T tamely ramified in K/k . If e is the ramification extension of index of P in K , we have e | q d − 1 where d = deg P . the rational function field The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 28 / 53
Tame ramification Gabriel Villa Salvador For the part (a), first we observe Introduction Cyclotomic function fields Proposici´ on The maximal abelian Let P ∈ R + T tamely ramified in K/k . If e is the ramification extension of index of P in K , we have e | q d − 1 where d = deg P . the rational function field The proof of The proof of this proposition is similar to that of the classical David Hayes Witt vectors case. and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 28 / 53
Tame ramification 2 Now we consider a tamely ramified abelian extension K/k where P 1 , . . . , P r are the finite prime divisors ramified in K/k . Gabriel Villa Let P ∈ { P 1 , . . . , P r } and with ramification index e . We Salvador consider k ⊆ E ⊆ k (Λ P ) with [ E : k ] = e . In E/k the prime Introduction divisor P has ramification e . Consider the composite KE . 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 Bibliography 29 / 53
Tame ramification 2 Now we consider a tamely ramified abelian extension K/k where P 1 , . . . , P r are the finite prime divisors ramified in K/k . Gabriel Villa Let P ∈ { P 1 , . . . , P r } and with ramification index e . We Salvador consider k ⊆ E ⊆ k (Λ P ) with [ E : k ] = e . In E/k the prime Introduction divisor P has ramification e . Consider the composite KE . Cyclotomic function fields K KE The maximal H abelian extension of the rational R function field The proof of David Hayes Witt vectors k E and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 29 / 53
Tame ramification 2 Now we consider a tamely ramified abelian extension K/k where P 1 , . . . , P r are the finite prime divisors ramified in K/k . Gabriel Villa Let P ∈ { P 1 , . . . , P r } and with ramification index e . We Salvador consider k ⊆ E ⊆ k (Λ P ) with [ E : k ] = e . In E/k the prime Introduction divisor P has ramification e . Consider the composite KE . Cyclotomic function fields K KE The maximal H abelian extension of the rational R function field The proof of David Hayes Witt vectors k E and the conductor From Abyankar’s Lemma we obtain that the ramification of P The in KE/k is e , so if we consider H , the inertia group of P in Kronecker– Weber–Hayes KE/k and R := ( KE ) H . Then P is unramified in R/k . Then Theorem Bibliography it can be proved that K ⊆ Rk (Λ P ) . 29 / 53
Proof of the tame ramification Gabriel Villa Salvador Continuing with this process r times we obtain that Introduction K ⊆ R 0 k (Λ P 1 ··· P r ) and where R 0 /k is an extension such that Cyclotomic the only possible ramified prime is p ∞ . Part (a) is consequence function fields of The maximal abelian extension of the rational function field The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 30 / 53
Proof of the tame ramification Gabriel Villa Salvador Continuing with this process r times we obtain that Introduction K ⊆ R 0 k (Λ P 1 ··· P r ) and where R 0 /k is an extension such that Cyclotomic the only possible ramified prime is p ∞ . Part (a) is consequence function fields of The maximal abelian extension of Proposici´ on the rational function field Let K/k be an abelian extension where at most a prime divisor The proof of David Hayes p of degree one is ramified and it is tamely ramified. Then K/k Witt vectors is an extension of constants. and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 30 / 53
Key fact: wild ramification Gabriel Villa Salvador Introduction Wild ramification is the key fact that distinguishes the positive Cyclotomic characteristic case from the classical one in the proof of the function fields Kronecker–Weber Theorem. In the classical case, the proof is The maximal abelian based in the fact that for p ≥ 3 , there is only one cyclic extension of the rational extension of degree p over Q where p is the only ramified function field prime. The case p = 2 is slightly harder since there are three The proof of David Hayes quadratic extensions where 2 is the only finite prime ramified. Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 31 / 53
Number fields vs function fields 1 Gabriel Villa In the function field case the situation is different. Fix a monic Salvador irreducible polynomial P ∈ R + T of degree d . Consider the Introduction Galois extension k (Λ P 2 ) /k . Then Gal( k (Λ P 2 ) /k ) = G P 2 . We Cyclotomic function fields have that G P 2 is isomorphic to the direct product of Gal( k (Λ P 2 ) /k ) = D P,P 2 with H := Gal( k (Λ P ) /k ) ∼ The maximal = C q d − 1 . abelian extension of the rational H function field F k (Λ P 2 ) The proof of David Hayes D P,P 2 D P,P 2 Witt vectors and the k H k (Λ P ) conductor The Kronecker– Weber–Hayes Theorem Bibliography 32 / 53
Number fields vs function fields 2 Gabriel Villa Salvador If F := k (Λ P 2 ) H , then Gal( F/k ) ∼ = D P,P 2 . Note that Introduction Cyclotomic = { A mod P 2 | A ∈ R T , A ≡ 1 mod P } D P,P 2 ∼ function fields The maximal abelian is an elementary abelian p –group so that D P,P 2 ∼ = C u p where extension of the rational u = sd , q = p s . In F/k the only ramified prime is P , it is function field wildly ramified and u can be as large as we want. This is one The proof of David Hayes of the reasons that the proof of the classical case using Witt vectors ramification groups seems not to be applicable here. and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 33 / 53
Main reduction step 1 Gabriel Villa Salvador Introduction We now study wild ramification. Thus, we have to show that if Cyclotomic L/k is a cyclic extension of degree p n for some n ∈ N we have function fields The maximal to show that L ⊆ F q pn k P α k m for some α, m ∈ N . abelian extension of The main simplification is given next on Witt generation of the rational function field cyclic extensions where we separate the ramification prime by The proof of prime. David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 34 / 53
Main reduction step 2 Gabriel Villa Theorem Salvador Let K/k be a cyclic extension of degree p n where Introduction P 1 , . . . , P r ∈ R + Cyclotomic T and possibly p ∞ , are the ramified prime function fields divisors. Then K = k ( � y ) where The maximal abelian extension of y p • y = � β = � • + � • • � − � δ 1 + · · · δ r + � µ, the rational function field The proof of with β p ∈ ℘ ( k ) , δ ij = Q ij 1 − β 1 / , e ij ≥ 0 , Q ij ∈ R T and if David Hayes eij P i Witt vectors e ij > 0 , then p ∤ e ij , gcd( Q ij , P i ) = 1 and and the deg( Q ij ) < deg( P e ij conductor ) , and µ j = f j ( T ) ∈ R T with p ∤ deg f j i The when f j �∈ F q . Kronecker– Weber–Hayes Theorem Bibliography 35 / 53
What remains to prove Cases (c) and (d) follow from (b) and the above theorem, so Gabriel the Kronecker–Weber Theorem will follow if we prove: 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 Kronecker– Weber–Hayes Theorem Bibliography 36 / 53
What remains to prove Cases (c) and (d) follow from (b) and the above theorem, so Gabriel the Kronecker–Weber Theorem will follow if we prove: Villa Salvador “Every cyclic extension K/k Introduction of degree p n where P ∈ R + Cyclotomic T is function fields The maximal the only ramified prime, P is abelian extension of the rational function field fully ramified and p ∞ is fully The proof of David Hayes decomposed, satisfies that Witt vectors and the conductor K ⊆ k P β = k (Λ P β ) for some The Kronecker– Weber–Hayes β ∈ N .” Theorem Bibliography 36 / 53
Elements of order p n in G P α Gabriel Villa Salvador Let P ∈ R + T , α ∈ N and let d := deg P . First we compute how many cyclic extensions of degree p n are contained in k (Λ P α ) . Introduction Cyclotomic Note that p ∞ is fully decomposed in K/k where K is any of function fields these extensions. The maximal abelian By direct computation we obtain that the number of elements extension of the rational of order p n in Gal( k (Λ P α ) /k ) is equal to function field The proof of � α α David Hayes � � � α ) − 1 � � d ( ) � − � q d ( α − p n − 1 p n q . (6.1) P n − 1 Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 37 / 53
Subgroups of order p n in G P α Gabriel Villa Salvador As a consequence we obtain Introduction Proposici´ on Cyclotomic function fields The number v n ( α ) of cyclic groups of order p n contained in The maximal abelian � ∗ is R T / ( P α ) extension of � the rational function field � α α α � � � � � ) − 1 The proof of d ( α − ) � d ( − p n p n − 1 p n − 1 � v n ( α ) = q q David Hayes . Witt vectors p n − 1 ( p − 1) and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 38 / 53
Artin–Schreier extensions Gabriel Villa Salvador Note that any K ⊆ k (Λ P α ) has conductor f K a divisor of P α . Next, we compute the number of cyclic extensions K of k of Introduction Cyclotomic degree p using the Theory of Artin–Schreier, such that P is the function fields only ramified prime, p ∞ decomposes and the conductor f K The maximal divides P α . Any such extension, written in normal form, is abelian extension of the rational given by an equation function field The proof of ℘y = y p − y = Q deg Q < deg P λ David Hayes p ∤ λ, P λ , λ > 0 , Witt vectors and the conductor and the conductor is f K = P λ +1 , so that λ ≤ α − 1 . The Kronecker– Weber–Hayes Theorem Bibliography 39 / 53
Number of Artin–Schreier extensions with given conductor and in normal form Now given another equation ℘z = z p − z = a written also in Gabriel normal form and such that k ( y ) = k ( z ) , satisfies that Villa a = j Q h Salvador P γ + ℘c with j ∈ { 1 , . . . , p − 1 } and c = P γ with pγ < λ . From these considerations, one may deduce that the Introduction number of different cyclic extensions K/k of degree p such Cyclotomic function fields � λ � λ − that the conductor K is f K = P λ +1 is equal to 1 p p − 1 Φ( P ) The maximal abelian where [ x ] denotes the integer function. So, the number of extension of the rational these extensions with conductor a divisor of P α is ω ( α ) p − 1 where function field The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 40 / 53
Number of Artin–Schreier extensions with given conductor and in normal form Now given another equation ℘z = z p − z = a written also in Gabriel normal form and such that k ( y ) = k ( z ) , satisfies that Villa a = j Q h Salvador P γ + ℘c with j ∈ { 1 , . . . , p − 1 } and c = P γ with pγ < λ . From these considerations, one may deduce that the Introduction number of different cyclic extensions K/k of degree p such Cyclotomic function fields � λ � λ − that the conductor K is f K = P λ +1 is equal to 1 p p − 1 Φ( P ) The maximal abelian where [ x ] denotes the integer function. So, the number of extension of the rational these extensions with conductor a divisor of P α is ω ( α ) p − 1 where function field The proof of David Hayes α − 1 � λ � λ − � Witt vectors p ω ( α ) = Φ( P ) . (6.2) and the conductor λ =1 gcd( λ,p )=1 The Kronecker– Weber–Hayes Theorem Bibliography 40 / 53
Number of Artin–Schreier extensions with given conductor and in normal form Now given another equation ℘z = z p − z = a written also in Gabriel normal form and such that k ( y ) = k ( z ) , satisfies that Villa a = j Q h Salvador P γ + ℘c with j ∈ { 1 , . . . , p − 1 } and c = P γ with pγ < λ . From these considerations, one may deduce that the Introduction number of different cyclic extensions K/k of degree p such Cyclotomic function fields � λ � λ − that the conductor K is f K = P λ +1 is equal to 1 p p − 1 Φ( P ) The maximal abelian where [ x ] denotes the integer function. So, the number of extension of the rational these extensions with conductor a divisor of P α is ω ( α ) p − 1 where function field The proof of David Hayes α − 1 � λ � λ − � Witt vectors p ω ( α ) = Φ( P ) . (6.2) and the conductor λ =1 gcd( λ,p )=1 The Kronecker– Weber–Hayes Computing (6.2) and comparing with last proposition we Theorem obtain ω ( α ) p − 1 = v 1 ( α ) . Bibliography 40 / 53
Case n = 1 Gabriel Villa Salvador In other words, every cyclic extensions K/k of degree p such Introduction that P is the only ramified prime, p ∞ decomposes fully in K/k and f K | P α is contained in k (Λ P α ) . Therefore the Cyclotomic function fields Kronecker–Weber Theorem holds in this case. The maximal abelian extension of the rational function field The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 41 / 53
Case n = 1 Gabriel Villa Salvador In other words, every cyclic extensions K/k of degree p such Introduction that P is the only ramified prime, p ∞ decomposes fully in K/k and f K | P α is contained in k (Λ P α ) . Therefore the Cyclotomic function fields Kronecker–Weber Theorem holds in this case. The maximal abelian Now we proceed with the cyclic case of degree p n . In other extension of the rational words, we want to prove that any cyclic extensions of degree p n function field of conductor a divisor P α and where p ∞ decomposes fully, is The proof of David Hayes contained in k (Λ P α ) . Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 41 / 53
Case n = 1 Gabriel Villa Salvador In other words, every cyclic extensions K/k of degree p such Introduction that P is the only ramified prime, p ∞ decomposes fully in K/k and f K | P α is contained in k (Λ P α ) . Therefore the Cyclotomic function fields Kronecker–Weber Theorem holds in this case. The maximal abelian Now we proceed with the cyclic case of degree p n . In other extension of the rational words, we want to prove that any cyclic extensions of degree p n function field of conductor a divisor P α and where p ∞ decomposes fully, is The proof of David Hayes contained in k (Λ P α ) . Witt vectors The proof is on induction on n . The case n = 1 is the case of and the conductor Artin–Schreier extensions. The Kronecker– Weber–Hayes Theorem Bibliography 41 / 53
Induction hypothesis Gabriel Villa Salvador We consider K n a cyclic extension of k of degree p n such that Introduction Cyclotomic P is the only ramified prime, P is fully ramified, p ∞ is fully function fields decomposed and f K n | P α . Let K n − 1 be the subfield of K n of The maximal abelian degree p n − 1 over k . Let K n /k be generated by the Witt vector extension of the rational y p − � � y = � β = ( β 1 , . . . , β n ) , that is, K n = k ( � y ) with ℘� y = � β function field and � β written is the normal form described by Schmid. Then The proof of David Hayes β ′ = ( β 1 , . . . , β n − 1 ) . K n − 1 /k is given by the Witt vector � Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 42 / 53
Case n − 1 Gabriel Let � Villa λ = ( λ 1 , . . . , λ n − 1 , λ n ) be the Schmid’s vector of Salvador invariants, that is, each β i is given by Introduction Cyclotomic β i = Q i function fields where Q i = 0 , that is, β i = 0 or P λ i The maximal abelian deg Q i < deg P λ i , gcd( Q i , P ) = 1 , extension of the rational λ i > 0 and gcd( λ i , p ) = 1 . function field The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 43 / 53
Case n − 1 Gabriel Let � Villa λ = ( λ 1 , . . . , λ n − 1 , λ n ) be the Schmid’s vector of Salvador invariants, that is, each β i is given by Introduction Cyclotomic β i = Q i function fields where Q i = 0 , that is, β i = 0 or P λ i The maximal abelian deg Q i < deg P λ i , gcd( Q i , P ) = 1 , extension of the rational λ i > 0 and gcd( λ i , p ) = 1 . function field The proof of David Hayes Since P is fully ramified, λ 1 > 0 . The next step is to find the Witt vectors number of different extensions K n /K n − 1 that can be and the conductor constructed by means of β n . If β n � = 0 , each equation in The normal form is given by Kronecker– Weber–Hayes Theorem Bibliography 43 / 53
Witt equation Gabriel Villa Salvador ℘y n = y p n − y n = z n − 1 + β n (6.3) Introduction Cyclotomic where z n − 1 is the element of K n − 1 obtained by the Witt’s function fields generation of K n − 1 with the vector � β ′ . In fact, formally, z n − 1 The maximal abelian is given by extension of the rational function field n − 1 1 The proof of � � p n − i � y p n − i + β p n − 1 � � z n − 1 = − y i + β i + z i − 1 David Hayes i i p n − 1 Witt vectors i =1 and the conductor with z 0 = 0 . The Kronecker– Weber–Hayes Theorem Bibliography 44 / 53
Case n − 1 second part As in the case n = 1 , we have that there exist at most Gabriel Villa � λ n � λ n − Salvador Φ( P p ) fields K n with λ n > 0 . The conductor of K n is P M n +1 with Introduction Cyclotomic M n = max { pM n − 1 , λ n } function fields and P M n − 1 +1 is the conductor of K n − 1 . The maximal abelian extension of the rational function field The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 45 / 53
Case n − 1 second part As in the case n = 1 , we have that there exist at most Gabriel Villa � λ n � λ n − Salvador Φ( P p ) fields K n with λ n > 0 . The conductor of K n is P M n +1 with Introduction Cyclotomic M n = max { pM n − 1 , λ n } function fields and P M n − 1 +1 is the conductor of K n − 1 . The maximal abelian extension of It follows that the rational function field pM n − 1 ≤ α − 1 , λ n ≤ α − 1 and The proof of David Hayes � α − 1 � Witt vectors f K n − 1 | P δ with δ = + 1 . and the p conductor The By the induction hypothesis, the number of such fields K n − 1 is Kronecker– Weber–Hayes v n − 1 ( δ ) . Theorem Bibliography 45 / 53
Case n Gabriel Villa Let t n ( α ) , n, α ∈ N be the number of cyclic extensions K n /k Salvador of degree p n with P the only ramified prime, fully ramified, p ∞ Introduction fully decomposed and f K n | P α . To prove the Cyclotomic function fields Kronecker–Weber Theorem it suffices to show t n ( α ) ≤ v n ( α ) . The maximal We have t 1 ( α ) = v 1 ( α ) = ω ( α ) p − 1 . By induction hypothesis we abelian extension of assume t n − 1 ( δ ) = v n − 1 ( δ ) . In general we have t n ( α ) ≥ v n ( α ) . the rational function field Now we obtain by direct computation The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 46 / 53
Case n Gabriel Villa Let t n ( α ) , n, α ∈ N be the number of cyclic extensions K n /k Salvador of degree p n with P the only ramified prime, fully ramified, p ∞ Introduction fully decomposed and f K n | P α . To prove the Cyclotomic function fields Kronecker–Weber Theorem it suffices to show t n ( α ) ≤ v n ( α ) . The maximal We have t 1 ( α ) = v 1 ( α ) = ω ( α ) p − 1 . By induction hypothesis we abelian extension of assume t n − 1 ( δ ) = v n − 1 ( δ ) . In general we have t n ( α ) ≥ v n ( α ) . the rational function field Now we obtain by direct computation The proof of David Hayes � α � d ( α − ) Witt vectors v n ( α ) v n ( δ ) = q p and the . (6.4) conductor p The Kronecker– Weber–Hayes Theorem Bibliography 46 / 53
Repetitions Gabriel Villa Salvador Considering the case β n = 0 , the number of fields K n Introduction containing a fixed field K n − 1 obtained in (6.2) is Cyclotomic function fields � α � d ( α − ) . The maximal 1 + ω ( α ) = q p abelian extension of the rational Finally, with the substitution y n �→ z := y n + jy 1 , function field j = 0 , 1 , . . . , p − 1 in (6.2) we obtain The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 47 / 53
Repetitions Gabriel Villa Salvador Considering the case β n = 0 , the number of fields K n Introduction containing a fixed field K n − 1 obtained in (6.2) is Cyclotomic function fields � α � d ( α − ) . The maximal 1 + ω ( α ) = q p abelian extension of the rational Finally, with the substitution y n �→ z := y n + jy 1 , function field j = 0 , 1 , . . . , p − 1 in (6.2) we obtain The proof of David Hayes ℘z = z p − z = β n + jβ 1 . Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 47 / 53
The miracle Gabriel Villa That is, each extension obtained in (6.2) is obtained p times Salvador or, equivalently, for each β n the same extension is obtained Introduction with β n , β n + β 1 , . . . , β n + ( p − 1) β 1 . It follows that for each Cyclotomic � α � ) of such d ( α − function fields K n − 1 there are at most 1+ ω ( α ) = 1 p p q p The maximal extensions K n . From equation (6.4) we obtain abelian extension of the rational function field The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 48 / 53
The miracle Gabriel Villa That is, each extension obtained in (6.2) is obtained p times Salvador or, equivalently, for each β n the same extension is obtained Introduction with β n , β n + β 1 , . . . , β n + ( p − 1) β 1 . It follows that for each Cyclotomic � α � ) of such d ( α − function fields K n − 1 there are at most 1+ ω ( α ) = 1 p p q p The maximal extensions K n . From equation (6.4) we obtain abelian extension of the rational � α � 1 � function field d ( α − ) � p t n ( α ) ≤ t n − 1 ( δ ) pq The proof of David Hayes � α � � 1 d ( α − ) � Witt vectors p = v n − 1 ( δ ) pq = v n ( α ) . and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 48 / 53
The miracle Gabriel Villa That is, each extension obtained in (6.2) is obtained p times Salvador or, equivalently, for each β n the same extension is obtained Introduction with β n , β n + β 1 , . . . , β n + ( p − 1) β 1 . It follows that for each Cyclotomic � α � ) of such d ( α − function fields K n − 1 there are at most 1+ ω ( α ) = 1 p p q p The maximal extensions K n . From equation (6.4) we obtain abelian extension of the rational � α � 1 � function field d ( α − ) � p t n ( α ) ≤ t n − 1 ( δ ) pq The proof of David Hayes � α � � 1 d ( α − ) � Witt vectors p = v n − 1 ( δ ) pq = v n ( α ) . and the conductor The This proves part (b) and the Theorem of Kronecker–Weber. Kronecker– Weber–Hayes Theorem Bibliography 48 / 53
References Gabriel Villa Carlitz, Leonard , On certain functions connected with Salvador polynomials in a Galois field , Duke Math. J. 1 (1935), Introduction 137–168. 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 Bibliography 49 / 53
References Gabriel Villa Carlitz, Leonard , On certain functions connected with Salvador polynomials in a Galois field , Duke Math. J. 1 (1935), Introduction 137–168. Cyclotomic function fields Carlitz, Leonard , A class of polynomials , Trans. The maximal Amer. Math. Soc. 43 (1938), 137–168. abelian extension of the rational function field The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 49 / 53
References Gabriel Villa Carlitz, Leonard , On certain functions connected with Salvador polynomials in a Galois field , Duke Math. J. 1 (1935), Introduction 137–168. Cyclotomic function fields Carlitz, Leonard , A class of polynomials , Trans. The maximal Amer. Math. Soc. 43 (1938), 137–168. abelian extension of the rational Hayes, David R. , Explicit Class Field Theory for function field Rational Function Fields , Trans. Amer. Math. Soc. 189 The proof of David Hayes (1974), 77–91. Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 49 / 53
References Gabriel Villa Carlitz, Leonard , On certain functions connected with Salvador polynomials in a Galois field , Duke Math. J. 1 (1935), Introduction 137–168. Cyclotomic function fields Carlitz, Leonard , A class of polynomials , Trans. The maximal Amer. Math. Soc. 43 (1938), 137–168. abelian extension of the rational Hayes, David R. , Explicit Class Field Theory for function field Rational Function Fields , Trans. Amer. Math. Soc. 189 The proof of David Hayes (1974), 77–91. Witt vectors and the Hilbert, David , Ein neuer Beweis des Kronecker’schen conductor Fundamentalsatzes ¨ uber Abel’sche Zahlk¨ orper , Nachr. Ges. The Kronecker– Wiss. zu Gottingen 1 (1896/97), 29–39. Weber–Hayes Theorem Bibliography 49 / 53
References 2 Gabriel Villa Kronecker, Leopold , ¨ Salvador Uber die algebraisch aufl¨ osbaren Gleichungen. I, Monatsber. Akad. Wiss. zu Berlin 1853, Introduction 356–374; II ibidem 1856, 203–215 = Werke, vol. 4 , Cyclotomic function fields Leipzig–Berlin 1929, 3–11, 27–37. The maximal abelian extension of the rational function field The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 50 / 53
References 2 Gabriel Villa Kronecker, Leopold , ¨ Salvador Uber die algebraisch aufl¨ osbaren Gleichungen. I, Monatsber. Akad. Wiss. zu Berlin 1853, Introduction 356–374; II ibidem 1856, 203–215 = Werke, vol. 4 , Cyclotomic function fields Leipzig–Berlin 1929, 3–11, 27–37. The maximal abelian Kronecker, Leopold , ¨ Uber Abelsche Gleichungen , extension of the rational Monatsber. Akad. Wiss. zu Berlin 1877, 845–851 = Werke, function field vol. 4 , Leipzig–Berlin 1929, 65–71. The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 50 / 53
References 2 Gabriel Villa Kronecker, Leopold , ¨ Salvador Uber die algebraisch aufl¨ osbaren Gleichungen. I, Monatsber. Akad. Wiss. zu Berlin 1853, Introduction 356–374; II ibidem 1856, 203–215 = Werke, vol. 4 , Cyclotomic function fields Leipzig–Berlin 1929, 3–11, 27–37. The maximal abelian Kronecker, Leopold , ¨ Uber Abelsche Gleichungen , extension of the rational Monatsber. Akad. Wiss. zu Berlin 1877, 845–851 = Werke, function field vol. 4 , Leipzig–Berlin 1929, 65–71. The proof of David Hayes Neumann, Olaf , Two proofs of the Kronecker–Weber Witt vectors and the theorem “according to Kronecker, and Weber” , J. Reine conductor Angew. Math. 323 (1981), 105–126. The Kronecker– Weber–Hayes Theorem Bibliography 50 / 53
References 3 Gabriel Villa Salvador Salas–Torres, Julio Cesar, Rzedowski–Calder´ Introduction on, Martha and Cyclotomic Villa–Salvador, Gabriel Daniel , Tamely ramified function fields extensions and cyclotomic fields in characteristic p , The maximal abelian Palestine Journal of Mathematics 2 (2013), 1–5. extension of the rational function field The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 51 / 53
References 3 Gabriel Villa Salvador Salas–Torres, Julio Cesar, Rzedowski–Calder´ Introduction on, Martha and Cyclotomic Villa–Salvador, Gabriel Daniel , Tamely ramified function fields extensions and cyclotomic fields in characteristic p , The maximal abelian Palestine Journal of Mathematics 2 (2013), 1–5. extension of the rational function field Salas–Torres, Julio Cesar, The proof of Rzedowski–Calder´ on, Martha and David Hayes Villa–Salvador, Gabriel Daniel , Artin–Schreier Witt vectors and the and Cyclotomic Extensions , to appear in JP Journal of conductor Algebra, Number Theory and Applications. The Kronecker– Weber–Hayes Theorem Bibliography 51 / 53
References 4 Gabriel Salas–Torres, Julio Cesar, Villa Rzedowski–Calder´ Salvador on, Martha and Villa–Salvador, Gabriel Daniel , A combinatorial Introduction proof of the Kronecker–Weber Theorem in positive Cyclotomic function fields characteristic , arXiv:1307.3590v1. The maximal abelian extension of the rational function field The proof of David Hayes Witt vectors and the conductor The Kronecker– Weber–Hayes Theorem Bibliography 52 / 53
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