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Nontrivial Galois module structure of cyclotomic fields. (English) Zbl 1020.11070

A tame Galois extension \(L/K\) with group \(G\) is said to have trivial Galois module structure if the projective \(O_K[G]\)-module \(O_L\) is free. A number field \(K\) is said to be Hilbert-Speiser iff all Abelian tame extensions \(L/K\) have trivial Galois module structure in this sense. The authors start from the result of D. Replogle, K. Rubin, A. Srivastav and the reviewer [J. Number Theory 79, 164-173 (1999; Zbl 0941.11044)] which says that \(\mathbb{Q}\) is the only Hilbert-Speiser field, and they take a closer look at the reasons that make every other number field \(K\) non-Hilbert-Speiser. In fact, for every prime number \(l\) one can attach to \(K\) the finite module \(V_l=(O_K/lO_K)^*/Im(O_K^*)\). Whenever there exists an odd prime \(l\) such that the exponent of \(V_l\) fails to divide \((l-1)^2/2\), the field \(K\) is not Hilbert-Speiser. Call \(l\) a witness for \(K\) if the non-divisibility just mentioned occurs for \(l\) and \(K\). In loc. cit., the required witnesses \(l\) for every \(K\) different from \({\mathbb{Q}}\) came from a fairly general Čebotarev argument. The authors propose to look for explicit values for \(l\), given a particular class of fields \(K\), to wit, the cyclotomic fields \(K_n={\mathbb{Q}}(\zeta_n)\). They use a nice mix of theory and computation. First, one may exclude all cases where \(K_n\) does not have class number 1, since then \(l=2\) is a witness (we haven’t explained what this means: it means \(V_2\) is nontrivial, which also suffices to make \(K\) a non-Hilbert-Speiser field). This only leaves a finite list of values for \(n\). It is very convenient that for all these \(n\), the units of \(K_n\) coincide with the cyclotomic units, which makes the numerical computations go rather smoothly. For each \(n\) in the finite list, the authors give a set of witnesses \(l\) in their Table 2.9. It is not stated clearly how exhaustive these sets are: there seems to be a cutoff at 100, and for instance \(l=3\) would work for \(n=25\) but is not given.
The reviewer would like to make the following comment: All fields treated in the paper under review are (totally) imaginary. This is in some sense the easy case for disproving the Hilbert-Speiser property, since one may look at the minus part of \(V_l\) (disregarding the 2-part), and this minus part is just the minus part of \((O_K/lO_K)^*\) factored out by the image of the roots of unity in \(K\), so it is much easier to control than \(V_l\) itself, for which one needs the unit group. In an early stage of the paper by Replogle, Rubin, Srivastav and the reviewer, there was the following result (with a 14-line proof): If \(K\) is an imaginary Abelian field which does not contain \(i=\sqrt{-1}\), then every \(l\equiv 3\) modulo 4 whose Frobenius in \(K\) is complex conjugation is a witness. For \(K=K_3\) (note \(n=3\) is on the finite list) this gives the witnesses \(l=11,23,47,59,71,83\) up to 100. (Interestingly, the set given in the paper under review is rather different: \(17,29,41,53,59,71,83,89\).) As a further illustration of the efficiency of this method, we look again at \(K={\mathbb{Q}}(\zeta_{25})\) and \(l=3\). Here \(l\) is totally inert in \(K\). Hence the minus part of \((O_K/3O_K)^*\) is cyclic of order \(3^{10}+1\). The odd part of this number is \(5^2\cdot 1181\), and \(K\) contains precisely the 50th roots of unity. Hence the minus part of the odd part of \(V_3\) is cyclic of order 1181, which is not a divisor of \((3-1)^2/2=2\), so \(l=3\) is a witness.
To sum up: The writing and the numerical computations in the paper under review are interesting and pleasant, but it seems that the potential of the theoretical approach has not been fully used yet. On the numerical side, it would be interesting to find explicit witnesses for the maximal real subfield of \(K_n\).

MSC:

11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R27 Units and factorization
11R29 Class numbers, class groups, discriminants
11R18 Cyclotomic extensions

Citations:

Zbl 0941.11044

Software:

SIMATH
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Full Text: DOI arXiv

References:

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