Kummer Type Extensions in Function Fields

International Journal of Algebra, Vol. 7, 2013, no. 4, 157-166 HIKARI Ltd, www.m-hikari.com Kummer Type Extensions in Function Fields Marco Sánchez-Mirafuentes Universidad Autónoma Metropolitana Unidad Iztapalapa División de Ciencias Básicas e Ingeniería Departamento de Matemáticas Av. San Rafael Atlixco 186, Col Vicentina Del. Iztapalapa, C.P. 09240 México, D.F. Gabriel Villa Salvador CINVESTAV IPN Departamento de Control Automático Apartado Postal 14-740 México, D.F. C.P 07000 Copyright c 2013 Marco Sánchez-Mirafuentes and Gabriel Villa Salvador. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We present a generalization of Kummer extensions in algebraic function fields with finite field of constants F q, using the action of Carlitz- Hayes. This generalization of Kummer type extensions is due to Wen- Chen Chi and Anly Li and due also to Fred Schultheis. The main results of this article are Proposition 3.2 and Theorem 3.4. They provide a partial analogue of a theorem of Kummer, which establishes a bijection between Kummer extensions L/K of exponent n and subgroups of K containing (K ) n. Mathematics Subject Classification: 11R60, 11R99, 12F05, 14H05 Keywords: Carlitz Module, Kummer extensions, Dual module, Bilinear map

158 M. Sánchez-Mirafuentes and G. Villa-Salvador 1 Introduction An n-kummer extension, is a field extension L/K such that n is relatively prime to char(k), the characteristic of K, μ n K, where μ n is the group of n-th roots of unity, and L = K( n Δ), where Δ is a subgroup of K containing the group K n of n-th powers, and K( n Δ) is the field generated by all the roots n a, with a Δ. This is the origin of the theory of Kummer which is of significance in class field theory, for example see [5] Chapter 1. In this setting we have the followings facts. Let K be any field containing the group μ n of n-th roots of unity, where n is a natural number prime to the characteristic of K. Then (1) An n-kummer extension L/K is a Galois extension and Gal(L/K) is an abelian group of exponent n. (2) If L/K is an abelian extension of exponent n, then L = K( n Δ), where Δ K and (K ) n Δ. We will present a generalization of Kummer extensions via the Carlitz- Hayes action. In what follows, p denotes a prime number, q = p s, s N, R T denotes the ring F q [T ], k denotes the field of rational functions F q (T ) and k denotes an algebraic closure of k. In Section 2 we present some facts over R T -modules that are used in Section 3. The main part of this work is precisely Section 3. We will discuss extensions L/K such that k K L k. Next we give a brief outline of the Carlitz Hayes action. For details see [2] and [7]. Let ϕ be the Frobenius automorphism ϕ : k k, ϕ(u) =u q, and let μ T be the homomorphism μ T : k k, μ T (u) =Tu. We have an action of R T in k given as follows: if M R T and u k, then u M := M(ϕ + μ T )(u). It can be shown that with this definition, k becomes an R T -module, see [7] Chapter 12. We have that z M is a separable polynomial in z of degree d = deg(m). Moreover, the polynomial z M can be written as z M = d i=0 (M,i)zqi, where (M,i) R T,(M,0) = M and (M,d) is the leader coefficient of M, see [7] Chapter 12. On the other hand, assuming that M R T \{0}, the M-torsion set of k, denoted by Λ M,isΛ M := {u k u M =0}. We also call Λ M to be the set of M-roots of Carlitz. Note that if a k then set of all roots of the polynomial z M a is {α + λ λ Λ M }, where α is any fixed root of z M a in k. Proposition 3.2 and Theorem 3.4 are analogous to facts (1) and (2) listed above, except that we only consider finite extensions.

Kummer type extensions in function fields 159 2 Module theory In this section, unless otherwise indicated, all modules and homomorphisms considered are R T -modules and R T -homomorphisms, respectively. Let A be a module, and a A. Let ϕ a : R T A be the homomorphism defined by ϕ a (M) =Ma. Definition 2.1. We say that A is a cyclic module, if there exists a A, such that ϕ a is a surjective homomorphism. Note that the above definition is equivalent to say that there exists a A such that A =(a). Let a A and consider the kernel of the homomorphism ϕ a, ker(ϕ a ). If ker(ϕ a ) {0} there exists a nonzero polynomial M that we may assume to be monic, such that ker(ϕ a )=(M). Definition 2.2. Let a A. We say that a has infinite order if the kernel of ϕ a is zero. We say that a has finite order M, where M is a monic polynomial, if ker(ϕ a ) is nonzero and ker(ϕ a )=(M). Now if A is a module, an exponent of A is a nonzero M R T, such that Ma = 0 for each a A. Remark 2.3. If A is a finite cyclic module, there exists a A such that ϕ a is a surjective homomorphism. Therefore there exists M R T \{0} such that (i) ker(ϕ a )=(M) and (ii) R T /(M) = A. As before, we may replace M by a monic polynomial and we say that A has order M. The proof of the following lemma is straightforward. Lemma 2.4. Let A be a cyclic module of order N, with N 0.LetN 1 be a monic divisor of N. Then there exists a submodule of A, of order N 1. Remark 2.5. With the conditions of Lemma 2.4, by Remark 2.3 we have Na =0. On the other hand, if B is a cyclic submodule of A of order N 1 then, again by Remark 2.3, there exists b B, such that ϕ b is surjective and ker(ϕ b )= (N 1 ). Since b A, there exists N 2 R T, such that b = N 2 a. Now, because Nb = N(N 2 a) = N 2 (Na) = 0 we have N ker(ϕ b ) = (N 1 ). Therefore N 1 is a divisor of N. Since all cyclic modules of order M are isomorphic to R T /(M) we have there is only one cyclic module of order M, it follows that for each monic divisor M of N, there is a unique cyclic submodule of A of order N.

160 M. Sánchez-Mirafuentes and G. Villa-Salvador Analogously to the case of cyclic groups C m, we denote by C M the module R T /(M) which is cyclic of order M. Definition 2.6. Let A be a module. We denote by  or by Hom R T (A, C M ), the group of homomorphisms from A into C M,  is the dual module of A. Let f : A B be a homomorphism, such that A and B have exponent M. Then we have a homomorphism f : B  defined by f(ψ) =ψ f. Note that this is a contravariant functor, namely () : R T -modules of exponent M R T -modules such that if f : A B and g : B C are homomorphisms, then (1) ĝ f = f ĝ and (2) 1 =1. Lemma 2.7. If A is a finite module, with exponent M such that A = B D, then  is isomorphic to B D. Proof. The natural projections π 1 : B D B and π 2 : B D D induce homomorphisms π 1 : B B D and π 2 : D B D. Therefore we may define θ : B D B D given by θ(ψ 1,ψ 2 ) = π 1 (ψ 1 )+ π 2 (ψ 2 ), where (ψ 1,ψ 2 ) B D. We have θ is a homomorphism. Moreover if ψ B D, since ψ is a homomorphism, we get ψ(x, y) =ψ(x, 0) + ψ(0,y) for all (x, y) B D. Now, if we define ψ 1 : B C M by ψ 1 (x) =ψ(x, 0) and ψ 2 : D C M by ψ 2 (y) =ψ(0,y) then ψ 1 and ψ 2 are homomorphisms. Thus, we obtain the induced map δ : B D B D given by δ(ψ) =(ψ 1,ψ 2 ) which is a module homomorphism whose inverse is θ. This completes the proof. Proposition 2.8. A finite module A with exponent M is isomorphic to its dual. In other words A =  Proof. By Theorem 4.7, Chapter 5 of [3], we can write A = P A P. The above sum is over all irreducible polynomials P and A P denotes the set of elements of A having order a power of P. Now by Theorem 4.9 of Chapter 5 of [3], each A P can be written as A P = C P α 1... C P α k, where α 1... α k 1, and each C P α i is a cyclic module whose generator has order P α i, so each C P α i has order P α i. Note that each A P and each C P α i have exponent M.

Kummer type extensions in function fields 161 By the above observation and by Lemma 2.7, we may assume that A is cyclic generated by a of order P α, where α N and P is irreducible. Hence the function ϕ a is an epimorphism and (P α )=ker(ϕ a ). Since M is an exponent of A, we have P α M. Now from Lemma 2.4 and Remark 2.3, C M has a unique cyclic submodule of order P α. We denote such module by C P α. The homomorphism ϕ a : R T A induces an isomorphism, denoted again by ϕ a : R T /(P α ) A. The inverse of the isomorphism ϕ a will be denoted by ψ. Let y = ψ(a). Then y is a generator of C P α. The composition ψ with the natural inclusion C P α C M, gives an element of Â, also denoted by ψ. Let ϕ Â. Note that Im(ϕ) C M is a cyclic submodule of order N. Thus, there exists w Im(ϕ) such that w = ϕ(a w ) where a w A generates Im(ϕ) and its order is N. On the other hand P α (N). Therefore P α = ND, for some D R T. Consequently N = P γ for some γ α, i.e., w has order P γ and as a w generates Im(ϕ), we have Im(ϕ) C P α. Now ϕ is determined by its action on a, where a A is a generator of A. Therefore ϕ(a) =Ny. Now if ψ N = Nψ then ψ N (a) =Nψ(a) =Ny = ϕ(a), that is, ϕ = ψ N (ψ). So  =(ψ) and the order of  is qdeg(p α). Therefore A = Â. Definition 2.9. Let A and B be modules. A bilinear map of A B into a module C, denoted by (a, b) < a,b>, is a function A B C that has the following property: for each a A, the function b < a,b>is a homomorphism and, for each b B, the function a < a,b>is a homomorphism. An element a A is said orthogonal to S B, if<a,b>= 0, for each b S. Analogously, we say that b B is orthogonal to S A if <a,b>= 0 for all a A. The left kernel of the bilinear function is the submodule of A that is orthogonal to B. The right kernel of the bilinear function is the submodule of B that is orthogonal to A. Let A and B be the left and right kernels of the bilinear map respectively. An element b B gives an element in Hom RT (A, C) given by a < a,b>, which we denote by ψ b. Then ψ b vanishes in A, i.e., ψ b (a) = 0 for each a A. Therefore, ψ b induces a homomorphism A/A C given by a + A ψ b (a). On the other hand, if b b mod B then ψ b = ψ b. Therefore we obtain firstly a homomorphism ψ : B/B Hom RT (A/A,C) given by ψ(b+b )=ψ b and secondly, an exact sequence of modules 0 B/B Hom RT (A/A,C). (1)

162 M. Sánchez-Mirafuentes and G. Villa-Salvador Similarly we obtain the exact sequence 0 A/A Hom RT (B/B,C). (2) Proposition 2.10. Let A A C M be a bilinear map of modules into a cyclic module C M of finite order M. Let B, B be their respective left and right kernels. Suppose that A /B is finite with exponent M and that A/B have exponent M. Then A/B is finite and A /B is isomorphic to the dual module of A/B. Proof. From the exact sequences (1) and (2) it follows that the following sequences are exact and 0 A /B Hom RT (A/B, C M ) (3) 0 A/B Hom RT (A /B,C M ). (4) From (4) we obtain that A/B can be viewed as a submodule of  /B. From this follows the finiteness of A/B. On the other hand we obtain from the exact sequences (3) and (4) and Proposition 2.8 and card(a/b) card( (A /B )) = card(a /B ). (5) card(a /B ) card( (A/B)) = card(a/b). (6) the second equality follows from Proposition 2.8. Now from (3) we have obtain injective map φ : A /B Â/B. Then by (5) and (6), φ is surjective, and this finishes the proof. 3 Kummer theory In this section we give a generalization of Kummer extensions, a little different from those given by Chi and Li [1] and Schultheis [6]. In what follows we will assume that the extensions considered are subextensions of k/k. Let M R T be a non-constant polynomial and ϕ : K K defined by ϕ(u) =u M, where K = k(λ M ). Then ϕ is a homomorphism. Moreover consider a submodule B of K under the action of Carlitz-Hayes containing K M = ϕ(k). M Let b K we shall use the symbol b to denote any such element β, which will be called an M-root of b. Since the M-roots of Carlitz are in K, we observe that field K(β) is the same no matter which M-root β of b we select. We denote this field by K( M b). For our next proposition we need the definition give in [1].

Kummer type extensions in function fields 163 Definition 3.1. An abelian extension L/K with Galois group G, is said to be an R T -abelian extension if its Galois group has an R T -module structure. An R T -abelian extension L/K is said to be of exponent M if G is an M-torsion R T -module, i.e., M σ = 1 for all σ G. Proposition 3.2. (1) Let B be an R T -submodule of K containing K M and let K B be the composite of all fields K( M b) with b B. Then K B /K is a Galois abelian extension. (2) Assume that K B /K is an R T -abelian extension of exponent M with Galois group G. Then there is a bilinear map G B Λ M given by (σ, a) < σ,a> where <σ,a>= σ(α) α and α M = a. The left kernel is 1 and the right kernel is K M. The extension K B /K es finite if only if (B : K M ) is finite. In this case we have B/K M = Ĝ In particular we have the equality [K B : K] =(B : K M ). Proof. (1) Let b B and let β be an M-root of b. The polynomial z M b splits into linear factors in K B for all b B. ThusK B /K is a Galois extension. Now let G = Gal(K B /K), σ G, b B and β a root of the polynomial z M b. Then σ(β) =β + λ Mσ for some M σ R T, where λ is a generator of Λ M. Therefore the map σ M σ is a injective homomorphism of G into Λ M. (2) We define G B Λ M by (σ, b) < σ,b>, where <σ,b>= σ(β) β and β M = b. This definition is independent of the choice of the M-root of b. We have <σ,a+ b>=< σ,a>+ <σ,b>for all a, b B and since the definition of <σ,b>is independent of the choice of the M-root of b, it follows <σ τ,b >=< σ,b>+ <τ,b>. Let σ G. Suppose <σ,a>= 0 for all a B. Then for every α of K B such that α M = a B we have σ(α) =α. Hence σ induces the identity on K B and the left kernel is 1. On the other hand, let a B and suppose <σ,a>= 0 for all σ G. Then σ(α) =α for all σ G. Therefore α K and a = α M K M. So the right kernel is K M. Now suppose that B/K M is finite. Since the right kernel is K M, from Proposition 2.10 we obtain that G = G/1 is finite. In particular K B /K is

164 M. Sánchez-Mirafuentes and G. Villa-Salvador finite. On the other hand if K B /K is finite then, from Proposition 2.10, we have that the sequence 0 B/K M Hom RT (G/1, Λ M ) is exact. Since Hom RT (G/1, Λ M ) is finite, it follows that B/K M is finite. Finally, since by Proposition 2.10 B/K M is isomorphic to the dual module of G, we have that B/K M = Ĝ,so[K B : K] =(B : K M ). For our next result we need definition given in [1] Definition 3.3. An R T -abelian extension L/K is said to be R T -cyclic if its Galois group is a cyclic R T -module. In this case, if Gal(L/K) = R T /(M), where M is a monic polynomial, we say that the R T -cyclic extension L/K is of order M. In the following theorem let M denote the set of R T -submodules which contain K M and let F denote the set of R T -abelian extensions of K with exponent M. Theorem 3.4. With the notation of Proposition 3.2, the function ϕ : M F given by ϕ(b) =K B is injective. Also if L/K is a finite R T -abelian extension finite of exponent M, then there is a submodule B of K containing K M, such that L = K B. Proof. To show that the function ϕ is injective, it is enough to prove that if K B1 K B2 then B 1 B 2, since from the equality ϕ(b 1 )=ϕ(b 2 ), it follows that K B1 K B2 and that K B2 K B1. Let b B 1. Then K( M b) K B2 and K( M b) is contained in a finitely generated subextension of K B2. Thus we may assume, without loss of generality, that B 2 /K M is finitely generated, hence finite. Now let β be such that β M = b. Let B 3 be the submodule of K generated by B 2 and b. We will show that K B2 = K B3. We have K B2 K B3. To show K B2 K B3, let α be any M-root of c B 3. Therefore c is of the form b N + s i=1 bn i i, with b i B 2. Then α M = b N + s i=1 bn i i = β MN + s i=1 βmn i i with βi M = b i, i =1,...,s, i.e., α = β N + s i=1 βn i i +λ A, where λ is a generator of Λ M. Therefore K(α) K B2. It follows that K B3 K B2. Thus by Proposition 3.2 (2) (B 2 : K M )=(B 3 : K M ). Hence b B 2,so that B 1 B 2. On the other hand, let K be a finite R T -abelian extension of K of exponent M. Let G = Gal(K /K). Then, by Theorem 4.7 and Theorem 4.9 Chapter 4 of [3], G is a finite direct sum of R T -submodules of exponent M. By Galois theory we may assume that the extension K /K is cyclic of exponent M. Now,

Kummer type extensions in function fields 165 by Proposition 2.6 of [1], we have that every R T -cyclic extension of exponent M is obtained by attaching an M-root of an element of K. Therefore there exits sets {b j } K, {α j } K such that αj M = b j and K = K({α j }). Let B be the submodule of K generated by {b j } and K M. Then K K B. On the other hand, consider an M-root of c B, sayα. So, α M = c. Other hand c = s j=1 bn j j + a M, a K. Then, we have α = s j=1 αn j j + a therefore K(α) K. It follows that K B K and ϕ(b) =K. This completes the proof. Proposition 3.5. Let L/K be a finite R T -abelian extension. Assume Λ N K, with N R T.Let W = {a = a + K N K/K N N a L}. Then W = Hom(G, Λ N ), where G = Gal(L/K). Proof. Let a W. We define a function ϕ a : G Λ N, given by ϕ a (σ) = σ(α) α, where α is an N-root of a, i.e., α N = a. The definition of ϕ a is independent of the root used. Note that ϕ a (σ τ) =σ(τ(α)) α = σ(τ(α) α + α) α = σ(τ(α) α)+σ(α) α = τ(α) α + σ(α) α. Thus ϕ a is a homomorphism of abelian groups. Therefore it is possible to define f : W Hom(G, Λ N ) given by f(α) =ϕ α. We have that f is a homomorphism of abelian groups. Now if f(a) =ϕ a =0 then σ(α) α = 0, for all σ G. In this way we obtain that α K. Since a = α N then a K N. In this way a =0,sof is injective. Now let ϕ : G Λ N be a homomorphism of abelian groups. Then ϕ(σ τ) =ϕ(σ)+ϕ(τ) =ϕ(σ)+σ(ϕ(τ)), that is, ϕ is a crossed homomorphism. By the additive Hilbert Theorem 90, there exists α L such that ϕ(σ) =σ(α) α. Therefore (σ(α) α) N = σ(α N ) α N =0. Thusa = α N K, and this shows that the function f is surjective. References [1] W-C. Chi, A. Li. Kummer theory of division points over Drinfeld modules of rank one, J. Pure Appl. Algebra, 156 (2001), 171-185.

166 M. Sánchez-Mirafuentes and G. Villa-Salvador [2] D.R. Hayes. Explicit class field theory for rational function fields, Trans. Amer. Math. Soc, 189 (1974), 77-91. [3] P. Hilton, Y. Wu. A course in modern algebra, Wiley Interscience Publication John Wiley and Sons. New York, 1974. [4] S. Lang. Algebra 3rd ed, Addison-Wesley Co. Reading, Mass, 1993. [5] J. Neukirch. Class field theory, Springer-Verlag, Berlin Heidelberg, 1986. [6] F. Schultheis. Carlitz-Kummer Function Fields, J. Number Theory, 36 (1990), 133-144. [7] G. D. Villa Salvador. Topics in the Theory of Algebraic Function Fields, Birkhäuser, Boston, 2006. Received: January, 2013