On the existence of Minkowski units in totally real cyclic fields

Transcription

1 Journal de Théorie des Nombres de Bordeaux 17 (2005), On the existence of Minkowski units in totally real cyclic fields ar František MARKO Résumé. Soit K un cors de nombres cyclique réel de degré n qui est le roduit de deux nombres remiers distincts et tel que le nombre de classes du n-ième cors cyclotomique soit égal à 1. Nous établissons certaines conditions nécessaires et suffisantes our l existence d une unité de Minkowski our K. Abstract. Let K be a totally real cyclic number field of degree n that is the roduct of two distinct rimes and such that the class number of the n-th cyclotomic field equals 1. We derive certain necessary and sufficient conditions for the existence of a Minkowski unit for K. 1. Introduction The main result (Theorem II.1) of [1] states a necessary and sufficient condition for the existence of the Minkowski unit in the totally real cyclic fields of degree n. These conditions are of an inductive nature as they relate the existence of the Minkowski unit in such a field K n to the existence of the Minkowski unit in the subfield K n 1. Additional necessary conditions are the isomorhism of the factor-grou of units E Kn /E Kn 1 and the grou Z[ζ n] of integers of the n -th cyclotomic field together with the surjectivity of the norm ma from K n to K n 1 on units modulo {±1}. For the sufficient conditions it is necessary to know the Minkowski unit of K n 1 and the generator of E Kn /E Kn 1 over Z[ζ n]. The relationshi between these generators is then described using a coordinate element from Z[ζ n]. If the coordinate element can be reresented by a unit of a certain ring and the reviously mentioned norm ma on units is surjective, the existence of the Minkowski unit follows. This method works best when all the invertible classes can be covered by units as is the case for the degrees n = 4, 8 and 9. For these degrees the existence of the Minkowski unit is equivalent to the surjectivity of the norm ma on units, see [1], Theorem II.2. The general idea of local generators and coordinates similar to [1] extends to totally real cyclic number fields of degree n for which the ring Z[ζ n ] of

2 196 František Marko the integers of the n-th cyclotomic field is a unique factorisation domain. We treat in detail the case when n is a roduct of two distinct rimes. Among other results we rove the following statement. Theorem 1.1. Let K be a totally real cyclic number field of degree n = 6, 10 or 14. Then K contains a Minkowski unit if and only if the norm mas from K to its subfields are surjective on units. For recise formulation see Corollary 5.3. This theorem is a generalization of Theorem 3 of [3]. 2. Notation Throughout the aer the symbols, q will denote distinct rime numbers, n a ositive comosite integer, K = K n a totally real cyclic number field of degree n over the field Q of rational numbers. Denote by σ a generator of the Galois grou G of K over Q, by K m a subfield of K of degree m > 1 over Q that is fixed by the subgrou of G generated by σ m. The letters m, l, d will be reserved for divisors of n that are larger than 1, hence they corresond to subfields of K. Let E Km be the grou of all units of the field K m and U Km = E Km /{±1}. Further denote by ϕ m (x) the m-th cyclotomic olynomial, let ζ m a rimitive m-th root of unity and ut ψ m (x) = n m 1 i=0 xim = xn 1 x m 1 = l n;l m ϕ l(x), O = Z[x]/( xn 1 x 1 ) and O m = Z[x]/(ϕ m (x)). Note that U K is not only a G-module but an O-module as well. A unit u E Km is called a Minkowski unit of K m if the conjugates of u generate U Km ; this is equivalent to the Galois module U Km being cyclic. For an O-module W denote W m = W ψm(x) and W m = m W m and W m = W m / W m. An O-module W will be called structured if Wm is a nontrivial cyclic O m -module for each m n. Actually W m = Om as O-modules in that case. The number of isomorhism classes of O-structured modules is denoted by M(n). We conjecture that each isomorhism class of structured O- modules contains U K for some number field K. In any case the index M(n) is an uer bound for the number of isoclasses of structured Galois modules U K for fields K of degree n. The function Ω(m) is defined as the sum of the exonents in the rime factorization of m, i.e. if m = t i=1 e i i, then Ω(m) = t i=1 e i. For a structured module W we say that its element w has length k and write l(w) = k if w m;ω(m)=k W m but w m;ω(m)=k 1 W m.

3 On Minkowski units in totally real cyclic fields Coordinates and Galois structure We start with an observation. l m;l m ϕ l (x). k Let m = s k=1 a k k, and ut A k(x) = Lemma 3.1. The ideal of Z[x] generated by olynomials A k for k = 1,..., s equals (ϕ m (x)). Proof. Clearly ϕ m (x) divides A k, so write A k = ϕ m (x)g k (x). The olynomials ϕ k are all irreducible, and this imlies that the olynomials g k are airwise relatively rime. Therefore the ideal of Z[x] generated by all g k s equals cz[x] for a certain non-zero c Z[x]. This shows that the ideal I generated by the A k s is rincial, and is generated by cϕ m (x). Hence all coefficients of olynomials lying in I are divisible by c, however the olynomals A k are monic, and this forces c to be 1. Existence of a Minkowski unit of K imlies certain structural roerties of the grous of units of subfields of K. Proosition 3.2. If K has a Minkowski unit, then Norm K/Km U K = U Km for each subfield K m of K. Moreover, the O-module U Km / m U K m is isomorhic to O m. Proof. If K has a Minkowski unit ɛ, then by [4], Pro.I.3, the Z[G]-module U K is isomorhic to O, where the generator ɛ corresonds to 1 and the conjugation by σ corresonds to multilication by x. Moreover, in this case according to [4], Pro.I.4, and its corollary, each subfield K m has a Minkowski unit and Norm K/Km U K = U Km for each subfield K m of K. The grou U Km is isomorhic to ψ m (x)o and the factor-grou U K / n U K n is isomorhic to O modulo the ideal generated by all ψ n (x). Using Lemma 3.1, we get that this factor-grou is isomorhic to Z[x]/(ϕ n (x)). It is an immediate consequence of R(m) that the ideal of O generated by all ψ m (x) for all m equals (ϕ m (x)ψ m (x)). Therefore for each m n the factor-grou U Km / m U K m is isomorhic to O m = Z[x]/(ϕ m (x)). Corollary 3.3. The O-module O is structured. Minkowski unit, then U = U K is structured. Moreover, if K has a Proof. Since U Km = Norm K/Km U = U ψm(x) we have U Km = U m for each m n. For secial values of n the Galois module U is structured if the norm mas on units are surjective (without the assumtion about the existence of the Minkowski unit). Proosition 3.4. If Z[ζ n ] is a rincial ideal domain and Norm K/Km U K = U Km for each subfield K m of K, then U K is structured.

4 198 František Marko Proof. According to [3], U K is isomorhic as a G-module to an ideal M of the ring O. Assuming that the norm ma on units are surjective (that is Norm K/Km U K = U Km for each subfield K m of K) we obtain that U m = U Km corresonds to module ψ m (x)m that is annihilated by l n;l m ϕ l(x) and Ũm is isomorhic to ψ m (x)m/(ψ m (x)m; m). Lemma 3.1 imlies that ψ m (x)m/(ψ m (x)m; m) is a module over Z[ζ m]. If Z[ζ n ] is a rincial ideal domain, then so is Z[ζ m ] and in this case Ũm is a cyclic Z[ζ m ]-module. The fact that it is nontrivial follows from the Dirichlet unit theorem. From now on assume that an O-module W is structured. Kee in mind the rominent examle of the structured O-module, namely the grou U K of the field K that is given by Proosition 3.4. Fix and denote by w m an element of W m whose class reresents a generator of Wm as an O m -module. Note that each w m can be written as w m = vm ψm(x) where v m W. Then each w W can be written uniquely as a roduct w = m n wαm(w) m, where α m (w) is a olynomial from O such that its degree does not exceed the degree of ϕ m (x). The exonents α m (u) are called coordinates of w. The elements {w m ; m d} (as well as {v m ; m d}) are generators of W as an O- module. In order to work with the above factorizations and to determine the O-structure of W, we will derive conditions describing when m n wam(x) m = 1 for arbritrary olynomials α m (x) from O. ψ d (x) ψm(x) For each w m and each d m we have wm = v ψ d(x) m. Denote a m,d = α d (v ψ d(x) m ). Then a m,m = 1 and the collection of all such a m,d for d m; d m is called the coordinate system of W. Proosition 3.5. Let a m (x); m n be olynomials in O. Then m n wam(x) m = 1 if and only if for all d n the condition C d : d m n n m a m(x)a m,d 0 (mod ϕ m (x)) is satisfied. Proof. We roceed in stes deending on the uer bound for the length of w = m n wam(x) m. First, l(w) < Ω(n) if and only if w W n which is equivalent to a n (x) 0 (mod ϕ n (x)) and that is condition C n. Next assume that l(w) k. We are looking for conditions that are equivalent to l(w) < k. The assumtion l(w) k means that w l n;l(l)=k W l, say w = l n;l(l)=k z l with z l W l. Choose l 0 n such that l(l 0 ) = k and consider w ψ l (x) 0. It is a roduct of z ψ l 0 (x) l 0 and other terms z ψ l 0 (x) l for l l 0. Since each z l = y ψ l(x) l for some y l W and ψ l0 (x)ψ l (x) (ψ l0 (x)ϕ l0 (x)) for l l 0, each z ψ lo (x) l belongs to W l 0. Therefore z l0 W l 0 if and only if w ψ l (x) 0 W l 0 which is the same as α l0 (w ψ l (x) 0 ) = 0. Since α l0 (w ψ l 0 (x) m ) = n l 0 a m,l0 the last

5 On Minkowski units in totally real cyclic fields 199 equality is equivalent to C l0. Therefore l(w) < k if and only if all conditions C l for l(l) = k are satisfied. Hence u = 1 if and only if all conditions C d for d n are satisfied. Although the above conditions C d do not describe exlicitly the Galois action on w m they rovide a means of checking whether the comutations involving Galois action on generators w m are correct. It is interesting that only the leading exonent a m,d in the factorization ψ d (x) ψm(x) of wm with resect to generators w l is needed in the above theorem. The olynomials a m,d are related by various congruences. They vary deending on the tye of decomosition of n into the roduct of rimes and therefore we will investigate the relationshi between a m,d searately for each such tye. In the next sections we will be looking for conditions on a coordinate system a m,d of U that will guarantee the existence of a Minkowski unit in K. For that urose we will investigate the relationshi between coordinate systems for various fields of the same degree n. 4. The case n = 2 The main riority in the case n = 2 is not to rove new results but to exlain our method in the simlest nontrivial case, show a connection with the work of [1], and view and rerove some of their results from a different ersective. First note that ϕ 2(x) (mod ϕ (x)). A structured O-module T is given by a generating set {t 2, t } and the coordinate a 2,. If t = t α 2 (t) t α(t) 2, then t ψ(x) = t a 2, α 2 (t)+α (t). Since T ψ(x) = T the congruence a 2,α 2(t)+α (t) 1 (mod ϕ (x)) must have a solution which is equivalent to a 2, being invertible modulo (, ϕ (x)). Conversely, if a olynomial a 2, is invertible modulo (, ϕ (x)), then we can define an O-module T generated by { t 2, t } that is structured and a 2, is its coordinate. Set T = O 2 O as an abelian grou and t 2 = (1, 0), t = (0, 1). A Z[x]-module structure on T is defined as c(x)(a(x), b(x)) = (f(x), a 2,e(x) + b(x)c(x)), where e(x) and f(x) are the quotient and the remainder resectively after the division of the olynomial a(x)c(x) by ϕ 2(x). Assume now that we have two structured O-modules T and W generated by {t 2, t } and {w 2, w } resectively, with coordinates a 2, and b 2, resectively. We would like to determine when T and W are isomorhic as O-modules. As a secial case, for number fields K and K of degree n = 4, 9 or 25 satisfying Norm K/K U K = U K and Norm U K /K = U K K this would allow us to comare Galois structures of U K and U K as well as to determine

6 200 František Marko whether K has an Minkowski unit (if U K is isomorhic to the structured module O). If L : T W is an O-morhism, then L(T ) W and therefore L(t 2) = w a(x) w b(x) 2 and L(t ) = w c(x) for some a(x) O and b(x), c(x) O. A ma L defined on generators in the above way is a morhism of O-modules if t k(x) t l(x) 2 = 1 for k(x) O, l(x) O imlies L(t 2) k(x) L(t ) l(x) = 1. Proosition 3.5 alied to T shows t k(x) t l(x) 2 = 1 if and only if k(x) 0 (mod ϕ 2(x)) and a 2,k(x) + l(x) 0 (mod ϕ (x)). That means k(x) = ϕ 2(x)k (x) and l(x) a 2,k (x) (mod ϕ (x)). Use Proosition 3.5 again for W to see that L(t 2) k(x) L(t ) l(x) = w a(x)k(x) w b(x)k(x)+c(x)l(x) 2 = 1 if and only if a(x)k(x) 0 (mod ϕ 2(x)) and b 2,a(x)k(x) + (b(x)k(x) + c(x)l(x)) 0 (mod ϕ (x)). The first congruence is clearly satisfied and the second one is equivalent to b 2,a(x)k (x) + b(x)k (x) c(x)a 2,k (x) 0 (mod ϕ (x)). Since there are no restrictions on the values of k (x) we can choose k (x) = 1 and obtain b 2,a(x) a 2,c(x) b(x) (mod ϕ (x)). This congruence gives a necessary and sufficient condition for L to be an O-morhism from T to W. Therefore the existence of an O-morhism L from T to W as above is given by the following congruence: b 2,a(x) a 2,c(x) (mod (, ϕ (x))). If this congruence is satisfied for some a(x), c(x), we can comute b(x) as required to make L an O-morhism. Assume now that L as above is an isomorhism. Then L(T ) = W and since L(t l(x) ) = w c(x)l(x), the class of c(x) modulo ϕ (x) must be invertible. Since L(T ) = W the class of a(x) modulo ϕ 2(x) must be invertible as well. Conversely, we will show that if a(x) is invertible modulo ϕ 2(x) and c(x) is invertible modulo ϕ (x), then L is an isomorhism. Assume t Ker(L) and k(x), l(x) are coordinates of t with resect to the generators {t 2, t }. Then L(t) = w a(x)k(x) w b(x)k(x)+c(x)l(x) 2 = 1 imlies a(x)k(x) 0 (mod ϕ 2(x)) and since the class of a(x) is invertible, we have k(x) = 0. Therefore L(t) = w c(x)l(x) = 1 and since the class of c(x) is invertible we

7 On Minkowski units in totally real cyclic fields 201 also have l(x) = 0 and thus t = 1. We have established that such ma L is injective. If c(x) is invertible modulo ϕ (x), then clearly L(T ) = W. If also the class of a(x) modulo ϕ 2(x) is invertible, then L also induces a ma onto W = W/W and thus L is surjective. Hence we determined the conditions for an isomorhism between structured modules. Proosition 4.1. Let T and W be structured O-modules for n = 2 with coordinates a 2, and b 2, resectively. Then T = W as O-modules if and only if there is a(x) O that is invertible modulo ϕ 2(x), c(x) O that is invertible modulo ϕ (x) and b 2,a(x) a 2,c(x) (mod (, ϕ (x))). When comaring this result with [1], Theorem II.1, it is worth noting that the congruence classes there aear as a consequence of a choice of generators whereas congruence classes here aear because of the requirement for morhism. The above criteria further simlifies for n = 2 = 4, 9 and 25. This is the case when the class number of Z[ζ n ] and Z[ζ ] equals 1 and hence all units in these rings are ± cyclotomic units. If a olynomial is reresented by a cyclotomic unit xa 1 for (a, ) = 1 modulo ϕ 2(x), then its class modulo ϕ (x) is also reresented by a unit. Thus if a(x) is reresented by a unit modulo ϕ 2(x) then a(x) is also reresented by a unit modulo ϕ (x). Therefore the condition of Proosition 4.1 is satisfied if and only if the invertible class of b 2,a 1 2, modulo (, ϕ (x)) is reresented by a unit modulo ϕ (x). This also means that if an isomorhism of T and W exists, then there is one with a(x) = 1. We have thus roved the following statement. Proosition 4.2. If n = 2 = 4, 9 or 25, then structured modules T and W are isomorhic as O-modules if and only if the invertible class of b 2,a 1 2, modulo (, ϕ (x)) is reresented by a unit modulo ϕ (x). Corollary 4.3. If K is a totally real cyclic number field of degree n = 2 = 4 or 9, then K contains a Minkowski unit if and only if Norm K/K U K = U K. Corollary 4.4. If K is a totally real cyclic number field of degree n = 25 such that Norm K/K5 U K = U K5, then K contains a Minkowski unit if and only if the class of its coordinate b 25,5 modulo (5, ϕ 5 (x)) is reresented by a unit modulo ϕ 5 (x). Proof. Assumtion Norm K/K U K = U K imlies that T = U K is structured. Put W = O in the above roosition. Since the coordinate b 2, = 1 for O, the modules U K and O are isomorhic (meaning a Minkowski unit x 1

8 202 František Marko exists for K) if and only if a 2, modulo (, ϕ (x)) has a reresentative that is a unit modulo ϕ (x). It can be checked easily that every for n = 4, 9, every invertible class modulo (, ϕ (x)) satisfies this condition. Corollary 4.5. M(25) = 5. Proof. It can be easily verified that the condition in Corollary 4.4 is true only for 20 out of 100 invertible classes. Therefore there are at most 5 nonisomorhic Galois structures of U K for fields K of degree n = 25 with Norm K/K5 U = U K5. 5. The case n = q We will roceed in the same way as outlined in section 4. First note that (ϕ (x), ϕ q (x)) = 1,ψ (x) = ϕ q (x)ϕ q (x) q (mod ϕ (x)) and ψ q (x) = ϕ q (x)ϕ (x) (mod ϕ q (x)). A structured O-module T is given by a generating set {t q, t, t q } and the coordinate system {a q,, a q,q }. If t = t αq(t) q t α(t) t αq(t) q, then t ψ(x) = t aq,αq(t)+qα(t) and t ψq(x) = t aq,qαq(t)+αq(t) q. Since T ψ(x) = T and T ψq(x) = T q the congruences a q, α q (t) + qα (t) 1 (mod ϕ (x)) and a q,q α q (t) + α q (t) 1 (mod ϕ q (x)) must have solutions which is equivalent to a q, being invertible modulo (q, ϕ (x)) and a q,q being invertible modulo (, ϕ (x)). Conversely, given a air {a q,, a q,q } such that the olynomial a q, is invertible modulo (q, ϕ (x)) and the olynomial a q,q is invertible modulo (, ϕ (x)), we can define an O-module T generated by { t q, t, t q } that is structured and {a q,, a q,q } is its coordinate system. Set T = O q O O q as an abelian grou and t q = (1, 0, 0), t = (0, 1, 0), t q = (0, 0, 1). Let γ (x) and γ q (x) be olynomials such that ϕ (x)γ (x) + ϕ q (x)γ q (x) = 1. A Z[x]-module structure on T is defined as d(x)(a(x), b(x), c(x)) = (f(x), γ (x)a q, e(x) + b(x)d(x), γ q (x)a q,q e(x) + c(x)d(x)), where e(x) and f(x) are the quotient and the remainder resectively after the division of the olynomial a(x)d(x) by ϕ q (x). Assume now that we have two structured O-modules T and W generated by {t q, t, t q } and {w q, w, w q } resectively with the coordinate systems {a q,, a q,q } and {b q,, b q,q } resectively. If L : T W is an O-morhism, then L(T ) W W q and therefore L(t q ) = w a(x) q w b(x) and L(T q ) wq c(x), L(t ) = w d(x) and L(t q ) = wq e(x) for some a(x) O, b(x), d(x) O and c(x), e(x) O q. A ma L defined on generators in the above way is a morhism of O-modules t l(x) t m(x) if t k(x) q L(t q ) k(x) L(t ) l(x) L(t q ) m(x) = 1. q = 1 for k(x) O, l(x) O and m(x) O q imlies

9 and and On Minkowski units in totally real cyclic fields 203 Proosition 3.5 alied to T shows t k(x) q t l(x) t m(x) q = 1 if and only if k(x) 0 (mod ϕ q (x)), a q, k(x) + ql(x) 0 (mod ϕ (x)) That means a q,q k(x) + m(x) 0 (mod ϕ q (x)). k(x) = ϕ q (x)k (x), l(x) a q, k (x) (mod ϕ (x)) m(x) a q,q k (x) Use Proosition 3.5 again for W to see that L(t q ) k(x) L(t ) l(x) L(t q ) m(x) = w a(x)k(x) q = 1 if and only if a(x)k(x) 0 (mod ϕ q (x)), and (mod ϕ q (x)). w b(x)k(x)+d(x)l(x) b q, a(x)k(x) + q(b(x)k(x) + d(x)l(x)) 0 b q,q a(x)k(x) + (c(x)k(x) + e(x)m(x)) 0 w c(x)k(x)+e(x)m(x) q (mod ϕ (x)) (mod ϕ q (x)). The first congruence is clearly satisfied and the later ones are equivalent to and b q, a(x)k (x) + qb(x)k (x) d(x)a q, k (x) 0 b q,q a(x)k (x) + c(x)k (x) e(x)a q,q k (x) 0 (mod ϕ (x)) (mod ϕ q (x)). Since there are no restrictions on the values of k (x) we can choose k (x) = 1 to verify that both and b q, a(x) a q, d(x) qb(x) b q,q a(x) a q,q e(x) c(x) (mod ϕ (x)) (mod ϕ q (x)) must have solutions. These congruences rovide a necessary and sufficient condition for L to be an O-morhism from T to W. If the following system of congruences b q, a(x) a q, d(x) b q,q a(x) a q,q e(x) (mod (q, ϕ (x)) (mod (, ϕ q (x)) has a solution, then we can comute b(x), c(x) as required to make L an O-morhism. An element t = t k(x) q t l(x) t m(x) q of T is been maed by L to w a(x)k(x) q w b(x)k(x)+d(x)l(x) w c(x)k(x)+e(x)m(x) q.

10 204 František Marko Assume now that L as above is an isomorhism. Then L(T ) = W and L(T q ) = W q and since L(t l(x) ) = w d(x)l(x) and L(t m(x) q ) = wq e(x)m(x) the classes of d(x) modulo ϕ (x) and e(x) modulo ϕ q (x) must be invertible. Since L(T ) = W the class of a(x) modulo ϕ q (x) must be invertible as well. Conversely, we will show that if a(x) is invertible modulo ϕ q (x), d(x) is invertible modulo ϕ (x) and e(x) is invertible modulo ϕ q (x), then L is an isomorhism. Assume t Ker(L) and k(x), l(x), m(x) are coordinates of t with resect to the generators {t q, t, t q }. Then L(t) = w a(x)k(x) q w b(x)k(x)+d(x)l(x) wq c(x)k(x)+e(x)m(x) = 1 imlies a(x)k(x) 0 (mod ϕ q (x)) and since the class of a(x) is invertible, we have k(x) = 0. Therefore L(t) = w d(x)l(x) wq e(x)m(x) = 1. Then (ϕ (x), ϕ q (x)) = 1 imlies w d(x)l(x) = 1 and w e(x)m(x) q = 1. Since the classes of d(x) and e(x) are invertible we must have l(x) = 0 and m(x) = 0 hence t = 1. We have established that such ma L is injective. If d(x) is invertible modulo ϕ (x) and e(x) is invertible modulo ϕ q (x), then clearly L(T ) = W and L(T q ) = W q. If also the class of a(x) modulo ϕ q (x) is invertible, then L also induces a ma onto W = W/W W q and hence L is surjective. Hence we determined the conditions for an isomorhism between structured modules. Proosition 5.1. Let T and W be structured O-modules for n = q with coordinate systems {a q,, a q,q } and {b q,, b q,q } resectively. Then T = W as O-modules if and only if there there is a(x) O that is invertible modulo ϕ q (x), d(x) O that is invertible modulo ϕ (x), e(x) O q that is invertible modulo ϕ q (x) and b q, a(x) a q, d(x) b q,q a(x) a q,q e(x) (mod (q, ϕ (x)) (mod (, ϕ q (x)). The above criteria further simlifies when the class number of Z[ζ q ] equals 1 and hence all units in Z[ζ q ], Z[ζ ] and Z[ζ q ] are ± cyclotomic units. This is the case if and only if n = 6, 10, 14, 15, 21, 22, 26, 33, 34, 35 or 38. We will look first at the case when n = 2, that is n = 6, 10, 14, 22, 26, 34, 38. If a(x) is invertible modulo ϕ 2 (x) then there is olynomial a (x) such that a(x)a (x) 1 (mod ϕ 2 (x)). Since ϕ 2 (x) = ϕ ( x) it is equivalent to a( x)a ( x) 1 (mod ϕ (x)) that is a( x) is invertible modulo ϕ (x). But a(x) a( x) (mod 2) shows that classes modulo (2, ϕ (x)) reresented by olynomials that are invertible modulo ϕ 2 (x) coincide with those classes modulo (2, ϕ (x)) reresented by olynomials invertible modulo

11 On Minkowski units in totally real cyclic fields 205 ϕ (x). We can therefore relace the first congruence of Proosition 5.1 by congruence b 2, a 1 2, d (x) (mod (2, ϕ (x)) for a olynomial d (x) invertible modulo ϕ (x). How about congruence classes of a(x) modulo (, x+1) for a(x) invertible modulo ϕ 2 (x)? For = 3, 5, 11, 13 and 19, the olynomial 1 x is invertible modulo ϕ 2 (x) and 1 x 2 (mod x + 1), and since 2 is a rimitive root modulo, every nonzero class modulo (, x + 1) is reresented by a olynomial invertible modulo ϕ 2 (x). For the remaining cases = 7 and 17, the olynomial 1 x+x 2 is invertible modulo ϕ 2 (x) and 1 x+x 2 3 (mod x + 1) and we use the fact that 3 is now a rimitive root modulo to see that each nonzero class modulo (, x+1) is reresented by a olynomial invertible modulo ϕ 2 (x). We have thus established that we can always choose a(x) such that the second congruence is satisfied. We have roved the following roosition. Proosition 5.2. Let n = 6, 10, 14, 22, 26, 34 or 38. Then the structured modules T and W are isomorhic as O-modules if and only if the invertible class of b 2, a 1 2, modulo (2, ϕ (x)) can be reresented by a olynomial invertible modulo ϕ (x). Corollary 5.3. If K is a totally real cyclic number field of degree n = 2 = 6, 10 or 14, then K contains a Minkowski unit if and only if Norm K/K U K = U K and Norm K/K2 U K = U K2. Corollary 5.4. If K is a totally real cyclic number field of degree n = 2 = 22, 26, 34 or 38 such that Norm K/K U K = U K and Norm K/K2 U K = U K2, then K contains a Minkowski unit if and only if the class of its coordinate b 2, modulo (2, ϕ (x)) is reresented by a unit modulo ϕ (x). Proof. Assumtion of the corollaries imly that T = U K is structured. For W = O the coordinate system a 2, = 1, a 2,2 = 1 and the modules U K and O are isomorhic (meaning Minkowski unit exists for K) if and only if the invertible class b 2, modulo (, ϕ (x)) has a reresentative that is a unit modulo ϕ (x). For n = 2 = 6, 10 owers of a cyclotomic unit 1 + x modulo ϕ (x) cover all nonzero classes modulo (2, ϕ (x)). For n = 2 = 14 the rime 2 has index 2 modulo 7 and hence ϕ 7 (x) modulo factors into a roduct of two irreducible olynomials of degree 3. Therefore there are 49 invertible classes modulo (2, ϕ 7 (x)) all of which can be reresented by classes that are roducts of owers of cyclotomic units 1 + x and 1 + x + x 2 modulo ϕ 7 (x). In relation to Corollary 5.4 it is natural to ask how many ossible classes b 2, modulo (2, ϕ (x)) are reresented by units modulo ϕ (x) or better yet what is the index of classes covered by units modulo ϕ (x) in the subrou of invertible classes modulo (2, ϕ (x))? This index coincides with M(n) for

12 206 František Marko these articular values of n. More generally, according to Proosition 5.1, the index M(n) equals the index of all coordinate systems {b q,, b q,q } for which b q, is reresented by a unit modulo ϕ (x) and b q,q is reresented by a unit modulo ϕ q (x) in the set of all coordinate systems {b q,, b q,q }. Proosition 5.5. M(22) = 3, M(26) = 5, M(34) = 17, M(38) = 27 and M(15) = 2, M(21) = 4, M(33) = 44, M(35) = 90. Proof. The claim can be established for n = 22, 26, 34 and 38 by using Corollary 5.4 and comuting the number of classes modulo (2, ϕ (x)) that can be covered by units modulo ϕ (x). For the remaining classes n = 15, 21, 33 and 35 we have to use Proosition 5.1. Also, in this case we used the generators for the n-th cyclotomic units from [2]. The comutation of these values was done using the software ackage PARI with the hel of my undergraduate student Anthony Disabella. Acknowledgement The author would like to thank the referee for a valuable suggestion. References [1] L. Bouvier, J. Payan Modules sur certains anneaux de Dedekind. J. Reine Angew. Math. 274/275 (1975), [2] R. Kučera On bases of the Stickelberger ideal and of the grou of circular units of a cyclotomic field. J. Number Theory 40 (1992), [3] F. Marko On the existence of -units and Minkowski units in totally real cyclic fields. Abh. Math. Sem. Univ. Hamburg 66 (1996), [4] N. Moser Unités et nombre de classes d une extension Galoisienne diédrale de Q. Abh. Math. Sem. Univ. Hamburg 48 (1979), František Marko Pennsylvania State University 76 University Drive Hazleton, PA 18202, USA and Mathematical Institute Slovak Academy of Sciences Štefánikova Bratislava, Slovakia fxm13su.edu