Computation of several cyclotomic Swan subgroups (Q2759103)

For \(O_K\) the ring of integers in any number field \(K\) and \(G\) any finite abelian group, let Sw\((O_K,G)\) (the Swan subgroup) denote the subgroup of Pic\((O_K[G])\) made up by the classes \([P]\) where \(P\) runs through the invertible \(O_K[G]\)-modules which become free upon base change with \(O_K[G] \to O_K \times O_K[G]/(N_G)\). There is a standard method of calculating this group via the relevant Mayer-Vietoris sequence. Nontriviality of this Swan subgroup for suitable \(K\) and \(G\) has been used to prove that \(K\) admits tame \(G\)-extensions without normal bases [see e.g. \textit{C. Greither}, \textit{D. R. Replogle}, \textit{K. Rubin}, and \textit{A. Srivastav}, J. Number Theory 79, 164-173 (1999; Zbl 0941.11044)]. In the paper at hand the authors consider a particularly simple case: \(G\) has two elements. Here the standard calculation easily yields that Sw\((O_K,G)\) is isomorphic to the quotient of the unit group \((O_K/2)^*\) by the image \(\overline E\) of the units of \(O_K\). If \(K={\mathbb Q}(\zeta_p)\) and \(K^+\) has class number 1, things simplify further since then \(\overline E=\overline C\) (with \(C\) the group of cyclotomic units of \(K\)), and the authors point out that with \(z=\overline{\zeta_p}\) two generators \(z\) and \(z+1\) suffice for \(\overline C\) if 2 is a primitive root modulo the odd prime \(p\) (equivalently: \(O_K/2\) is a field). NEWLINENEWLINENEWLINEUsing a MAPLE package written by the first author, the above-mentioned quotient is calculated for just ten values of \(p\), the largest being \(p=61\), and the quotient turns out to be nontrivial in all examples with \(p>5\). Unfortunately the authors say nothing about the algorithms. It appears that there exist more efficient packages. For instance, the reviewer determined in a few seconds of computing time that for the next value \(p=67\) the order of \(z+1\) is \(575525617597=(2^{33}-1)*67\), using PARI. Quite generally \((2^{(p-1)/2}-1)\cdot p\) is the largest possible value, see 2) below. The reviewer also ran PARI for \(p=87\), which was almost as fast and gave the result \((2^{41}-1)\cdot 87\) again. On the theoretical side, the authors have overlooked two important facts: NEWLINENEWLINENEWLINE1) The image \(\overline C\) of the Galois module generated by the cyclotomic units is generated by \(z+1\) alone: if we let \(c\) (complex conjugation) act, we get \(z = (z+1)^{1-c}\). The authors did observe that this holds in their ten examples, see first sentence after the table on p. 347. NEWLINENEWLINENEWLINE2) If one only wants the quotient \((O_K/2)^*/\overline C\) to be nontrivial, no calculation is necessary at all. One knows that \(C=\langle \zeta_p\rangle C^+\) with \(C^+\) the group of cyclotomic units of \(K^+\). The image of \(C^+\) is in \(U:=(O_{K^+}/2)^*\). Thus if we quotient by \(U\), the image of \(C\) in \((O_K/2)^*/U\) has at most order \(p\). Under the assumption that \(O_K/2\) is a field, \((O_K/2)^*/U\) has order \(2^{(p-1)/2}+1\), which exceeds \(p\) for \(p>5\). Thus one gets much more than stated in Corollary 3. This also explains why the numbers in the right hand column of the table on p. 347 must grow fairly rapidly.NEWLINENEWLINENEWLINEIt appears that the problem of calculating Swan groups would become more substantial for totally real fields \(K\); that problem would lend itself naturally to computer calculations.