International Journal of Algebra, Vol. 7, 2013, no. 4, 157 - 166 HIKARI Ltd, www.m-hikari.com Kummer Type Extensions in Function Fields Marco S´anchez-Mirafuentes Universidad Aut´onoma Metropolitana Unidad Iztapalapa Divisi´on de Ciencias B´asicas e Ingenier´ia Departamento de Matem´aticas Av. San Rafael Atlixco 186, Col Vicentina Del. Iztapalapa, C.P. 09240 M´exico, D.F. Gabriel Villa Salvador CINVESTAV IPN Departamento de Control Autom´atico Apartado Postal 14-740 M´exico, D.F. C.P 07000 Copyright c 2013 Marco S´anchez-Mirafuentes and Gabriel Villa Salvador. This is an open access article distributed under the Creative Commons Attribution License, which per- mits 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 func- tion fields with finite field of constants Fq, 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. S´anchez-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, μn ⊆ K, where μn is the group of n-th roots of unity, and √ L = K( n Δ), ∗ ∗n where√ Δ is a subgroup of K containing the group√ K 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 μn 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- s Hayes action. In what follows, p denotes a prime number, q = p , s ∈ N, RT denotes the ring Fq[T ], k denotes the field of rational functions Fq(T ) and k denotes an algebraic closure of k. In Section 2 we present some facts over RT -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]. q Let ϕ be the Frobenius automorphism ϕ : k → k, ϕ(u)=u , and let μT be the homomorphism μT : k → k, μT (u)=Tu. We have an action of RT in M k given as follows: if M ∈ RT and u ∈ k, then u := M(ϕ + μT )(u). It can be shown that with this definition, k becomes an RT -module, see [7] Chapter M 12. We have that z is a separable polynomial in z of degree d = deg(M). M M d qi Moreover, the polynomial z can be written as z = i=0(M,i)z , where (M,i) ∈ RT ,(M,0) = M and (M,d) is the leader coefficient of M, see [7] Chapter 12. On the other hand, assuming that M ∈ RT \{0}, the M-torsion set of k, M denoted by ΛM ,isΛM := {u ∈ k | u =0}. We also call ΛM to be the set of M-roots of Carlitz. Note that if a ∈ k then set of all roots of the polynomial M M z − a is {α + λ | λ ∈ ΛM }, where α is any fixed root of z − 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 RT -modules and RT -homomorphisms, respectively. Let A be a module, and a ∈ A. Let ϕa : RT → A be the homomorphism defined by ϕa(M)=Ma. Definition 2.1. We say that A is a cyclic module, if there exists a ∈ A, such that ϕa 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 ϕa, ker(ϕa). If ker(ϕa) = {0} there exists a nonzero polynomial M that we may assume to be monic, such that ker(ϕa)=(M). Definition 2.2. Let a ∈ A. We say that a has infinite order if the kernel of ϕa is zero. We say that a has finite order M, where M is a monic polynomial, if ker(ϕa) is nonzero and ker(ϕa)=(M). Now if A is a module, an exponent of A is a nonzero M ∈ RT , such that Ma = 0 for each a ∈ A. Remark 2.3. If A is a finite cyclic module, there exists a ∈ A such that ϕa is a surjective homomorphism. Therefore there exists M ∈ RT \{0} such that (i) ker(ϕa)=(M) and ∼ (ii) RT /(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 .LetN1 be a monic divisor of N. Then there exists a submodule of A, of order N1. 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 N1 then, again by Remark 2.3, there exists b ∈ B, such that ϕb is surjective and ker(ϕb)= (N1). Since b ∈ A, there exists N2 ∈ RT , such that b = N2a. Now, because Nb = N(N2a)=N2(Na) = 0 we have N ∈ ker(ϕb)= (N1). Therefore N1 is a divisor of N. Since all cyclic modules of order M are isomorphic to RT /(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. S´anchez-Mirafuentes and G. Villa-Salvador Analogously to the case of cyclic groups Cm, we denote by CM the module RT /(M) which is cyclic of order M. Definition 2.6. Let A be a module. We denote by A or by HomRT (A, CM ), the group of homomorphisms from A into CM , A 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 → A defined by f(ψ)=ψ ◦ f. Note that this is a contravariant functor, namely () : RT -modules of exponent M → RT -modules such that if f : A → B and g : B → C are homomorphisms, then (1) g◦ f = f◦ g and (2) 1=1. Lemma 2.7. If A is a finite module, with exponent M such that A = B×D, then A is isomorphic to B × D. Proof. The natural projections π1 : B × D → B and π2 : B × D → D induce homomorphisms π1 : B → B × D and π2 : D → B × D. Therefore we may define θ : B × D → B × D given by θ(ψ1,ψ2)=π1(ψ1)+π2(ψ2), where (ψ1,ψ2) ∈ 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 ψ1 : B → CM by ψ1(x)=ψ(x, 0) and ψ2 : D → CM by ψ2(y)=ψ(0,y) then ψ1 and ψ2 are homomorphisms. Thus, we obtain the induced map δ : B× D → B × D given by δ(ψ)=(ψ1,ψ2) 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 = A ∼ Proof. By Theorem 4.7, Chapter 5 of [3], we can write A = P AP . The above sum is over all irreducible polynomials P and AP denotes the set of elements of A having order a power of P . ∼ Now by Theorem 4.9 of Chapter 5 of [3], each AP can be written as AP = CP α1 ⊕ ...⊕ CP αk , where α1 ≥ ...≥ αk ≥ 1, and each CP αi is a cyclic module αi αi whose generator has order P , so each CP αi has order P . Note that each AP and each CP αi 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 α the function ϕa is an epimorphism and (P )=ker(ϕa).