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Corrigendum to "Real abelian fields satisfying the Hilbert-Speiser condition for some small primes p".

Proc. Japan Acad. Ser. A Math. Sci. 92 (2016), no. 1, 19 22.

We use the same notation as in the paper [7]. In particular, for a prime number p, a number field F satisfies the Hilbert-Speiser condition (Hp) when every tame cyclic extension K/F of degree p has a normal integral basis. In [7], we claimed the following two results.

Proposition 1. Let p [greater than or equal to] 7 be a prime number with p [equivalent to] 3 mod 4. Let F be a number field unramified at p, and let N = F([square root of -p]). If F satisfies the Hilbert-Speiser condition ([H.sub.p]), then the exponent of the class group [Cl.sub.N] of N divides h(Q([square root of -p])).

Proposition 2. Let p [greater than or equal to] 7 be a prime number with h(Q([square root of -p])) = 1. When p = 7 (resp. 11), a real abelian field F satisfies ([H.sub.p]) if and only if F = Q([square root of 5]) or Q([square root of 13]) (resp. F = Q(cos2[pi]/7)). When p = 19, 43, 67 or 163, there is no real abelian field satisfying ([H.sub.p]).

In Proposition 2, we are excluding the case F = Q because the rationals Q satisfies ([H.sub.p]) for all p.

In his email of 30th May 2018, Fabio Ferri kindly informed us that the formula [[A.sub.[DELTA]] : [S.sub.[DELTA]] = [h.sup.-.sub.k] in [7, eq (2)] is incorrect and provided a counterexample. As he pointed out, the mistake was caused by our confusion of the ideal [S.sub.[DELTA]] with the Stickelberger ideal associated to Q([square root of -p]) by Sinnott [12]. In [7], we proved Proposition 1 using the incorrect formula, and proved Proposition 2 using Proposition 1. We could not confirm whether or not the assertions of Proposition 1 and its corollary ([7, Corollary]) are true. However, we can save the situation by replacing Proposition 1 with the following weaker assertion on the minus class group [Cl.sup.-.sub.N] of N = F([square root of -p]). Here, [Cl.sup.-.sub.N] is defined to be the kernel of the norm map [Cl.sub.N] [right arrow] [Cl.sub.F].

Proposition 3. Let p [greater than or equal to] 7 be a prime number with p = 3 mod 4, and let F be a totally real number field satisfying the Hilbert-Speiser condition [(.sub.Hp]). Then the exponent of the minus class group [Cl.sup.-.sub.N] of the CM-field N = F ([square root of -p]) divides h(Q([square root of -p])), and the exponent of [Cl.sub.F] divides (p - 1)/2.

Proposition 2 is correct. In the following, we show Proposition 3, and change and correct the proof of Proposition 2 in [7] using Proposition 3. In the proof of Proposition 3, we partially repeat some of the arguments in [7] for the convenience of the reader.

Proof of Proposition 3. Let G = [(Z/pZ).sup.x] be the multiplicative group, which we naturally identify with the Galois group Gal(Q([[zeta].sub.p])/Q). We define elements [[theta].sub.G] and [[theta].sub.2] of Q[G] by

[[theta].sub.G] = 1/p [p-1.summation over (a=1)] a[[sigma].sup.-1.sub.a] and [[theta].sub.2] = (2 - [[sigma].sub.2])[[theta].sub.G]

where [[sigma].sub.a] = a mod p is the automorphism of Q([[zeta].sub.p]) sending [[zeta].sub.p] to [[zeta].sup.a.sub.p]. The Stickelberger ideal [S.sub.G] of the group ring Z[G] is defined by

[S.sub.G] = Z[G][intersection][[theta].sub.G]Z[G],

We have p[[theta].sub.G] [member of] [S.sub.G] by the definition of [S.sub.G], and [[theta].sub.2] [member of] [S.sub.G] by [13, Lemma 6.9].

In this paragraph, let F denote an arbitrary number field. Let [GAMMA] = Z/pZ be the additive group. Denote by Cl([O.sub.F][[GAMMA]]) the locally free class group associated to the group ring [O.sub.F][[GAMMA]], and by [Cl.sup.0]([O.sub.F][[GAMMA]]) the kernel of the map Cl([O.sub.F][[GAMMA]]) [right arrow] [Cl.sub.F] induced by the augmentation [O.sub.F][[GAMMA]] [right arrow] [O.sub.F].

Through the natural action of G = [(Z/pZ).sup.x] on [GAMMA], the groups Cl([O.sub.F][[GAMMA]]) and [Cl.sub.0]([O.sub.F][[GAMMA]]) are regarded as modules over the group ring Z[G]. By the main theorem of McCulloh [9], F satisfies ([H.sub.p]) if and only if the condition

(1) [Cl.sub.0]([O.sub.F][[GAMMA]])[S.sub.G] = {0} is satisfied.

Now, let p and F be as in Proposition 3. Let N = F([square root of -p]) and let K = F([[zeta].sub.p]). Note that N is contained in K since p [equivalent to] 3 mod 4. As F satisfies ([H.sub.p]) and p [greater than or equal to] 7, the extension F/Q is unramified at p by Greither and Johnston [4, Theorem 1.1]. Hence, the Galois group Gal(K/F) naturally identifies with G = Gal(Q([[zeta].sub.p])/Q) via restriction. Let [mathematical expression not reproducible] be the ray class group of K defined modulo the ideal [[bar.w].sub.p][O.sub.K] with [[bar.w].sub.p] = [[zeta].sub.p] - 1. The class groups [mathematical expression not reproducible] and [Cl.sub.K] are regarded as modules over Z[G] by the above identification. As F/Q is unramified at p, it follows that [mathematical expression not reproducible] as Z[G]-modules by Brinkhuis [1, Proposition 2.1]. Therefore, we see from (1) that the Stickelberger ideal [S.sub.G] annihilates [mathematical expression not reproducible] and [Cl.sub.K]. It follows that [S.sub.G] annihilates [Cl.sub.N] (resp. [Cl.sub.F]) since the norm map from [Cl.sub.K] to [Cl.sub.N] (resp. [Cl.sub.F]) is surjective by [13, Theorem 10.1].

We denote by [chi] the quadratic character of G = [(Z/pZ).sup.x], and we extend it to a ring homomorphism Z[G] [right arrow] Z by linearlity. The restriction of the automorphism [[sigma].sub.a] [member of] G to Q([square root of -p]) and N = F([square root of -p]) is the trivial map or the complex conjugation depending on whether [chi](a) = 1 or -1, respectively. Accordingly, [[sigma].sub.a] acts on the minus class group [Cl.sup.-.sub.N] trivially or via (-1)-multiplication. This implies that [alpha] [member of] Z[G] acts on [Cl.sup.-.sub.N] via [chi]([alpha])-multiplication. Here recall the following class number formula (see (6.2) of Frohlich and Taylor [6, Chapter VIII]):

(2) h(Q([square root of -p])) = - 1/p [p-1.summation over (a=1)][a.sub.[chi]](a).

We already know that the elements -p[[theta].sub.G] and -[[theta].sub.2] belong to [S.sub.G] and hence they annihilate [Cl.sup.-.sub.N]. By (2), we observe that [chi](-p[[theta].sub.g]) = ph(Q([square root of -p])) and that

[chi](-[[theta].sub.2]) = (2 - [chi](2))[chi](-[[theta].sub.G])

equals h(Q([square root of -p])) or 3h(Q([square root of -p])) depending on whether [chi](2) = 1 or -1, respectively. Now we see that h(Q([square root of -p]))-multiplication annihilates [Cl.sup.-.sub.N] as p [greater than or equal to] 7.

Let [[chi].sub.0] be the trivial character of G = [(Z/pZ).sup.x], which extends to a ring homomorphism Z[G] [right arrow] Z by linearlity. As [[sigma].sub.a] [member of] G acts on [Cl.sub.F] trivially, the element [[theta].sub.2] [member of] [S.sub.G] acts on [Cl.sub.F] via multiplication by [[chi].sub.0]([[theta].sub.2]) = (p - 1)/2. We obtain the assertion for [Cl.sub.F] because [S.sub.G] annihilates [Cl.sub.F].

Corrected proof of Proposition 2. Let p [greater than or equal to] 7 be an odd prime number with h(Q([square root of -p])) = 1. Let F [not equal to] Q be a real abelian field satisfying [(.sub.Hp]), and let d = [F : Q] and N = F([square root of -p]). Then F/Q is unramified at p by [4, Theorem 1.1], and [h.sup.-.sub.N] = 1 by Proposition 3. Imaginary abelian fields K with [h.sup.-.sub.K] = 1 are determined by Louboutin [8], Park and Kwon [10,11] and Chang and Kwon [2,3]. In our setting where K = N = F([square root of -p]), we have the following three cases:

(I) d = 3,

(II) d [greater than or equal to] 5 and N/Q is a cyclic extension,

(III) N/Q is non-cyclic.

The fields F and Q([square root of -p]) are linearly disjoint over Q as F/Q is unramified at p. Therefore, d is odd for case (II), and conversely, the case where d is even is contained in (III). Case (I) is dealt with in [10], case (II) in [2], and case (III) in [3].

First, let us deal with case (I) under the notation in [10]. All imaginary sectic fields K with relative class number 1 are listed in [10,Table 3]. The fields K are parametrized with the conductors f of K, [f.sup.+] of [K.sup.+] and m of the imaginary quadratic subfield of K. In our

case K = N = F([square root of -p]), we have m = p and p [??] [f.sup.+] as F/Q is unramified at p. From the table, we find that F/Q is unramified at p and [h.sup.-.sub.N] = 1 when and only when (i) p = 7 and F is the cyclic cubic field of conductor 9 or 13 or (ii) p = 11 and F is the cyclic cubic field of conductor 7.

Next, let us deal with case (II) under the notation in [2]. All imaginary cyclic fields K such that [K : Q] [greater than or equal to] 10, [K : Q] is not a 2-power and [h.sup.-.sub.K] = 1 are listed in [2, Table I]. Among them we need those ones with [K : Q]/2 is odd, namely those ones in the upper half of the table. This is because d is odd for case (II). Such fields K are parametrized with the conductors [f.sub.K] of K, [mathematical expression not reproducible] of [K.sup.+] and [f.sub.2] of the imaginary quadratic subfield of K. In our case K = F([square root of -p]), we have [f.sub.2] = p and [mathematical expression not reproducible]. In the table, we find no such fields.

Finally, let us deal with case (III) under the notation in [3]. All imaginary non-cyclic fields K with relative class number 1 are listed in [3, Table I]. The table is arranged according to the type of the Galois group G = Gal(K/Q). Let us look at those ones with type G =(2*, 2*). These are imaginary (2, 2)-extensions of Q. They are parametrized with the conductors [mathematical expression not reproducible] of the imaginary quadratic subfields [k.sub.1] and [k.sub.2] of K. Then, in our case K = N = F([square root of -p]), we have [f.sub.1] = p, p|[f.sub.2] and p [??] [f.sub.2]/p (swapping [f.sub.1] and [f.sub.2] if necessary). From the table, we find F/Q is unramified at p and [h.sup.-.sub.N] = 1 when and only when (iii) p = 7 and F = Q([square root of 5]), Q([square root of 13]) or Q([square root of 61]) or (iv) p = 11 and F = Q([square root of 2]) or Q([square root of 17]). Next let us look at those K with G = (2*, 2*, 2*). These are imaginary (2, 2, 2)-extension. They are parametrized by conductors of three imaginary quadratic subfields similary to the case G = (2*, 2*). From the table, we find no desired pair (p, F). Now let us look at those K with G [not equal to] (2*, 2*), (2*, 2*, 2*). These K are parametrized with a set of generators of the group [X.sub.K] of the associated Dirichlet characters. In our case K = F([square root of -p]), [X.sub.K] contains [[chi].sup.(p-1)/2.sub.p] where [[chi].sub.p] is a Dirichlet character of conductor p and order p - 1. For each K in the table, we checked that p is ramified in [K.sup.+] from the data on [X.sub.K]. Therefore, we obtain no desired pair (p, F) from those K.

Therefore, we obtain 8 pairs (p, F) such that F/Q is unramified at p and [h.sup.-.sub.N] = 1, namely those listed in (i) (iv) above. Fortunately, these 8 pairs coincide with the pairs which we dealt with in [7]. We have already shown in [7] that ([H.sub.p]) is satisfied when p = 7 (resp. 11) and F = Q([square root of 5]) or Q([square root of 13]) (resp. Q(cos2[pi]/7)), and that ([H.sub.p]) is not satisfied for the other 5 pairs. Thus the proof of Proposition 2 is corrected.

Remark 1. Let p be as in Proposition 2, and let F be a real abelian field satisfying ([H.sub.p]). Then, Proposition 1 asserts [h.sub.N] = 1, while Proposition 3 asserts that [h.sub.N] = 1. So what we have actually determined in [7] using Proposition 1 is all real abelian fields F [not equal to] Q satisfying ([H.sub.p]) and [h.sub.N] = 1.

Remark 2. A correct proof of Proposition 2 is also given in F. Ferri and C. Greither [5, [section]6]. It is slightly different from ours. The subject of [5] is a "[C.sub.p]-Leopoldt field", a number field satisfying a condition somewhat weaker than ([H.sub.p]). They obtained several conditions for a field F to be [C.sub.p]-Leopoldt using the main theorem of [9]. One of them asserts that if h(Q([square root of -p])) = 1, then a real abelian field F unramified at p is [C.sub.p]-Leopoldt only when [h.sup.-.sub.N] = 1 with N = F([square root of -p]). Again, they show that [h.sup.-N] = 1 is a necessary condition for ([H.sub.p]) to hold. Using the above cited papers on relative class numbers, they observed that there is no F [not equal to] Q unramified at p such that [h.sup.-.sub.N] = 1 but [h.sub.+.sub.N] > 1. This gives an alternative proof of Proposition 2 (see Remark 1).

doi: 10.3792/pjaa.95.80

Acknowledgements. The author heartfully thanks Mr. Fabio Ferri for kindly informing him of the serious error contained in the previous paper [7]. The author is very grateful to the anonymous referee for several useful comments and for encouraging him to rewrite the first version of this corrigendum, thanks to which he was able to improve the presentation of the whole paper.

References

[1] J. Brinkhuis, Normal integral bases and complex conjugation, J. Reine Angew. Math. 375/376 (1987), 157 166.

[2] K.-Y. Chang and S.-H. Kwon, Class number problem for imaginary cyclic number fields, J. Number Theory 73 (1998), no. 2, 318-338.

[3] K.-Y. Chang and S.-H. Kwon, Class numbers of imaginary abelian number fields, Proc. Amer. Math. Soc. 128 (2000), no. 9, 2517-2528.

[4] C. Greither and H. Johnston, On totally real Hilbert-Speiser fields of type [C.sub.p], Acta Arith. 138 (2009), no. 4, 329-336.

[5] F. Ferri and C. Greither, Tame Galois module structure revisited, Ann. Mat. Pura Appl. (2019), doi: 10.1007/s10231-019-00852-x.

[6] A. Frohlich and M. J. Taylor, Algebraic number theory, Cambridge Studies in Advanced Mathematics, 27, Cambridge University Press, Cambridge, 1993.

[7] H. Ichimura, Real abelian fields satisfying the Hilbert-Speiser condition for some small primes p, Proc. Japan Acad. Ser. A Math. Sci. 92 (2016), no. 1, 19-22.

[8] S. Louboutin, CM-fields with cyclic ideal class groups of 2-power orders, J. Number Theory 67 (1997), no. 1, 1-10.

[9] L. R. McCulloh, Galois module structure of elementary abelian extensions, J. Algebra 82 (1983), no. 1, 102-134.

[10] Y.-H. Park and S.-H. Kwon, Determination of all imaginary abelian sextic number fields with class number < 11, Acta Arith. 82 (1997), no. 1, 27-43.

[11] Y.-H. Park and S.-H. Kwon, Determination of all non-quadratic imaginary cyclic number fields of 2-power degree with relative class number [less than or equal to] 20, Acta Arith. 83 (1998), no. 3, 211-223.

[12] W. Sinnott, On the Stickelberger ideal and the circular units of an abelian field, Invent. Math. 62 (1980/81), no. 2, 181-234.

[13] L. C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, 83, Springer-Verlag, New York, 1997.

By Humio ICHIMURA

Faculty of Science, Ibaraki University, Bunkyo 2-1-1 Mito, Ibaraki 310-8512, Japan

(Communicated by Shigefumi Mori, m.j.a., June 12, 2019)
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