On metacyclic extensions

Transcription

1 On metacyclic extensions Masanari Kida 1 Introduction A group G is called metacyclic if it contains a normal cyclic subgroup N such that the quotient group G/N is also cyclic. The category of metacyclic groups contains important families of groups such as dihedral groups and Frobenius groups. A Galois extension L over a field k is called a metacyclic extension if the Galois group Gal(L/k is isomorphic to a metacyclic group. In this paper, we give a method to construct metacyclic extensions using a Kummer theory without roots of unity studied in our previous paper [4]. Évariste Galois already knew that if a polynomial of prime degree l( 5 is solvable, then the Galois group of the polynomial is a Frobenius group (see [3, Lemma 7.1.2]. Thus our method enables us to construct solvable extensions. In their paper [5], Nakano and Sase also study a construction of metacyclic extensions, which is a generalization of a result of Imaoka and Kishi [2] on Frobenius and dihedral extensions. Our construction supersedes their construction in some respect. Furthermore, because our construction comes from a Kummer theory, we can expect to get more algebraic and arithmetic information about the extensions. The following notations are used throughout this paper. Let l be an odd prime number fixed once for all. We also fix a base field k whose characteristic is different from l. We shall impose some conditions on k later. We assume that any separable extensions of k are contained in a fixed separable closure k s of k. For an integer m prime to l, we denote by ζ m a primitive m-th root of unity in k s. For any separable extension K over k, we write K c for the l-th cyclotomic extension K(ζ l. Japan-Korea Joint Seminar on Number Theory and Related Topics 2008 held at Tohoku University from November 12 to 15, This research is supported in part by Grant-in-Aid for Scientific Research (C (No , Ministry of Education, Science, Sports and Culture, Japan. 1

2 2 Preliminaries In this section, we give some preliminaries from group theory and a Kummer theory without root of unity studied in [4]. 2.1 Metacyclic groups We refer to [1, Section 47] for basic facts on metacyclic groups. A metacyclic group G is an extension of a cyclic group with cyclic kernel N. In this paper, we only consider the case where the order of N is the prime number l. Then we can obtain the following description of G: M l (r s = a, b a l = 1, b r = 1, b 1 ab = a x, ord(x mod l = s, where we have It is easy to observe that s r l 1. (1 M l (r s is abelian s = 1, M l (r s is a Frobenius group F lr s = r, M l (r s is a dihedral group D 2l s = r = 2. Note that this notation for metacyclic groups is slightly different from one used in [5]. 2.2 Kummer theory via algebraic tori We denote n = [k c : k]. Suppose that there exists an integer-coefficient polynomial P(t = c 1 + c 2 t + + c n t n 1 Z[t] (2 of degree n 1 satisfying the following two conditions: and Z[ζ n ]/(P(ζ n Z/(l; (3 P(ζ i n Z[ζ i n] for all i with (n, i > 1, (4 where ζ n is a fixed primitive n-th root of unity. We further assume that the ring isomorphism (3 induces a group isomorphism ϕ k : Gal(k c /k ζ n mod P(ζ n. (5 2

3 Let R kc /kg m be the Weil restriction of scalars of the multiplicative group. The algebraic torus R kc /kg m is an n-dimensional torus defined over k whose splitting field is k c. Then the circulant matrix c 1 c 2 c n c n c 1 c n 1 circ(c 1, c 2,..., c n =... c 2 c 3 c 1 defines an endomorphism of degree l on the character module of R kc /kg m. In fact, we can show that det(circ(c 1, c 2,..., c n = l under our assumptions. In the dual category, we have a self-isogeny of degree l on the k-torus R kc /kg m. The following theorem is proved in [4]. Theorem 2.1. Let λ be the self-isogeny of R kc /kg m of degree l defined in the above. Then every point in the kernel of λ is k-rational and the exact sequence attached to the isogeny λ 1 ker λ R kc /kg m λ R kc /kg m 1 (exact induces the Kummer duality κ k : R kc /kg m (k/λr kc /kg m (k Hom cont (Gal(k s /k, ker λ(k. (6 As a consequence, every cyclic extension over k of degree l is of the form k(λ 1 (P with some P R kc /kg m (k. It is shown that this Kummer theory holds for the following cases: If n is a prime and if there exists an element λ Z[ζ n ] whose norm is l. Then we can find P(t satisfying our assumptions. We can always construct P(t in the case where n = 4. The base field of the Kummer theory can be descended to the field Q of rationals numbers when l = 3, 5, 7, 11. See [4, Section 4] for the detail. 3 Construction of metacyclic extensions We start from the base field k. We assume that there exists a Kummer theory explained in the preceding section over k for the prime l. We take a divisor d of l 1 and let F/k be a cyclic extension of degree d 3

4 satisfying F k c = k. Our aim is constructing all metacyclic extensions over k with given intermediate field F. By our assumption, the isomorphism (6 naturally lifts to the isomorphism κ F : R Fc /FG m (F/λR Fc /FG m (F Hom cont (Gal(k s /F, ker λ(k. (7 The isomorphism (7 classifies the cyclic extensions over F. We use this isomorphism to construct metacyclic extensions L over k that are cyclic over F. For our purpose, we regard as R Fc /FG m (F = R F/k (R Fc /FG m (k and decompose the k-torus R F/k (R Fc /FG m = R Fc /kg m by an isogeny. Let σ be a generator of the Galois group of F/k and τ a generator of the Galois group of F c /F: Gal(F/k = σ, Gal(F c /F = τ. We denote the degree of the cyclotomic extension F c /F by n. Since F k c = k by assumption, we have an isomorphism Let Gal(F c /k σ τ. RFc /kg m be the character module. Then we have R Fc /kg m Z Q = Q[Gal(F c /k] = Q[ σ τ ] = Q[ σ ] Q Q[ τ ]. We define a complex character χ of Gal(F/k by For j = 1, 2,..., d, let χ : Gal(F/k C, χ(σ = ζ d. e j = 1 d 1 χ j (σ i σ i Q(ζ l 1 [σ]. d i=0 These e j s are orthogonal idempotents of Q(ζ l 1 [σ]. Now, for any positive divisor s of d, we set ε s = e j, ord(ζ j d =s where the sum is taken over the integers j (modulo d such that the order of ζ j d is s, namely over j satisfying s = d/ gcd(j, d. We say that e j belongs 4

5 to ε s in this situation. Then it is easy to see that ε s Q[σ] for all s and they are, indeed, orthogonal idempotents of Q[σ]. We have a decomposition of R Fc /kg m Q: R Fc /kg m Q = (ε s Q[σ] Q[τ]. s d Let R(s be an algebraic k-torus whose characteristic module is the Z-span of a Q-basis of ε s Q[σ] Q Q[τ], which is determined up to isogeny. Now we have an isogeny ι defined over k ι : R(s R F/k (R Fc /FG m (8 s d of degree dividing some power of d. In particular, the degree is prime to l. The isogeny λ naturally induces an isogeny on R(s, for which we use the same notation λ, and we have an injective homomorphism: R(s(k/λR(s(k R Fc /kg m (k/λr Fc /kg m (k R Fc /FG m (F/λR Fc /FG m (F Hom cont (Gal(k s /F, ker λ(k. For a k-rational point P R(s(k, let P F denote the image of P in R Fc /FG m (F. Now we regard e j as an element of the l-adic group algebra Z l [σ] and reduce it modulo l to obtain the idempotents e j F l [σ]. We remark here that d is prime to l and, since the degree of λ is l, the group Hom cont (Gal(k s /F, ker λ(k is a F l -vector space. Therefore F l [σ] acts on R(s(k/λ s R(s(k and e j s decompose this quotient. Our main theorem is the following. Theorem 3.1. Suppose that e j belongs to ε s. If P e j (R(s(k/λR(s(k, then k(λ 1 (P F is a Galois extension with Galois group isomorphic to M l (d s. Moreover every M l (d s-extension over k containing the cyclic extension F appears in this manner. The proof of this theorem will appear elsewhere. 4 Examples In this section, we give explicit examples of Theorem 3.1 for small l. 5

6 Example 4.1. Let l = 3 and k = Q. The polynomial P(t = 2 t defines an isogeny of R Qc /QG m over Q of degree 3. Let F/Q be a quadratic extension different from Q( 3. We have a Kummer duality R Fc /FG m (F/λR Fc /FG m (F Hom cont (Gal(Q s /F, ker λ(q. (9 On the other hand, we have an isogeny ι in (8. In this case, the torus R(2 can be written down explicitly: ι : R Qc /QG m ker(n Fc /Q c : R Fc /QG m R Qc /QG m R Fc /QG m, whose inverse isogeny (in the sense of [6, Proposition 1.3.1] is given by ( x 2 x N Fc /Q c (x,. N Fc /Q c (x The corresponding group algebra decomposition is ( 1 σ Q[Gal(F c /Q] Q[Gal(F c /Q] 2 ( 1 + σ 2 Q[Gal(F c /Q]. We note that we obtain the inverse isogeny of ι by clearing denominators in the group algebra isomorphism. Let P F R Fc /FG m (F and L P = F(λ 1 (P F. The extension L P /F is a cubic cyclic extension. Our result implies that, when we write P F = ι(p, P R Qc /QG m (Q = L P /Q is a C 6 -extension; P ker ( N Fc /Q c (Q = LP /Q is a D 3 -extension. Example 4.2. Let l = 5 and k = Q. We take a cyclic extension F over Q degree dividing 4 that is disjoint from Q c. Then circ(1, 1, 1, 0 induces a Kummer duality of the same form as (9. If [F : Q] = 4, then ( ι : R Qc /QG m ker N F σ 2 : R c /Q c F σ 2 G c /Q m R Qc /QG m ( ker N Fc /F σ2 c Similarly as in the previous example, we have : R Fc /QG m R F σ 2 G c /Q m P R Qc /QG m (Q = L P /Q is a C 20 -extension; ( P ker N F σ 2 (Q = L c /Q P /Q is an M 5 (4 2-extension; c 6 R Fc /QG m.

7 ( P ker N Fc /F σ2 c in the same notation as above. If [F : Q] = 2, the isogeny is (Q = L P /Q is an F 20 -extension ι : R Qc /QG m ker(n Fc /Q c : R Fc /QG m R Qc /QG m R Fc /QG m and we have P R Qc /QG m (Q = L P /Q is a C 10 -extension; P ker ( N Fc /Q c (Q = LP /Q is a D 5 -extension. References [1] C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers, a division of John Wiley & Sons, New York-London, [2] M. Imaoka and Y. Kishi, On dihedral extensions and Frobenius extensions, Galois theory and modular forms, Dev. Math., vol. 11, Kluwer Acad. Publ., Boston, MA, 2004, pp [3] C. U. Jensen, A. Ledet, and N. Yui, Generic polynomials, Mathematical Sciences Research Institute Publications, vol. 45, Cambridge University Press, Cambridge, [4] M. Kida, Descent Kummer theory via Weil restriction of multiplicative groups, Preprint (2008. [5] S. Nakano and M. Sase, A note on the construction of metacyclic extensions, Tokyo J. Math. 25 (2002, no. 1, [6] T. Ono, Arithmetic of algebraic tori, Ann. of Math. (2 74 (1961, Masanari Kida Department of Mathematics, University of Electro-Communications, Chofugaoka 1-5-1, Chofu Tokyo Japan kida@sugaku.e-one.uec.ac.jp 7