Kummer's lemma for some cyclotomic fields (Q1382860)

Let \(K_f\) denote the cyclotomic field with conductor \(f\) and let \(p\) be an odd prime. When is a unit \(\epsilon\in K_f\) a \(p\)th power in \(K_f\)? If \(f=p\), the following condition guarantees that this is the case: \(\epsilon\) is congruent to a rational integer mod \(p^{M+1}\), where \(M\) is the highest exponent of \(p\) occurring in the prime factorizations of the numbers \(L_p(1,\chi)\) for even nonprincipal characters \(\chi\) of \(K_p\). Here \(L_p(s,\chi)\) denotes the \(p\)-adic \(L\)-function. This result was proved by L. C. Washington; it is a generalization of a classical theorem, known as Kummer's lemma (in which \(M=0\), that is, \(p\) is regular). In a subsequent article [J. Number Theory 40, 165-173 (1992; Zbl 0746.11043)] \textit{L. C. Washington} provided an extension of his result to the case \(f=p^n\), \(n\geq 1\). The present author finds another extension. He takes \(f=mp\), where \(p\) does not divide \(m\varphi (m)\) and \(p\) is congruent to 1 modulo the exponent of Gal\((K_m/\mathbb{Q})\) (\(\varphi\) denotes Euler's function). His proof uses the same idea but requires a more elaborate calculation. A crucial ingredient is the well-known explicit expression for \(L_p(1,\chi)\).