On a normal integral basis problem over cyclotomic \(Z_{p}\)-extensions. II.
Let \(p\) be an odd prime, \(K\) an imaginary abelian field containing a \(p\)th root of unity and \(K_\infty/K\) the cyclotomic \(\mathbb Z_p\)-extension with its \(n\)th layer \(K_n\). In the previous paper [J. Math. Soc. Japan 48, 689--703 (1996; Zbl 0892.11036)] the author proved that, under some assumptions on \(K\), for any unramified Kummer extension \(L/K_n\) of degree \(p\), \(LK_{n+1}/K_{n+1}\) does have a normal integral basis (NIB) even if \(L/K_n\) has no NIB. In the present paper, the author considers this type of problem for a tamely ramified Kummer extension \(L/K_n\) of degree \(p\). For a technical reason, the author studies only those extensions whose Kummer generator is square-free and is congruent to 1 modulo each prime ideal of \(K_n\) over \(p\). The paper contains many results which are too involved and technical to be formulated here. The special case of a biquadratic field \(K\) with \(p=3\) not splitting in \(K\) is nicely described in Proposition 2.
scientific article; zbMATH DE number 1886711