Units of irregular cyclotomic fields (Q5904221)

Let \(B_n\) denote the \(n\)-th Bernoulli number. For even \(i\) with \(2\le i\le p-3\), \textit{P. Dénes} [Publ. Math. 3, 17--23 (1954; Zbl 0056.03301); ibid. 3, 195--204 (1954; Zbl 0058.26902); ibid. 4, 163--170 (1956; Zbl 0071.26505)] defined \(u_i\) to be the smallest \(j\ge 0\) such that \(B_{ip^j}\not\equiv 0 \pmod{p^{2j +1}}\). Under the assumption that \(u_i\) is finite, he proved a number of results on units of cyclotomic fields. His proofs used Kummer's logarithmic differential quotient and involved some rather complicated calculations. In the present paper, it is shown that \(u_i = v_p(L_p(1;\omega^i))\), where \(L_p\) is the \(p\)-adic \(L\)-function and \(\omega\) is the Teichmüller character. Since \(L_p(1;\omega^i)\ne 0\) by a result of Ax and Brumer, \(u_i\) is finite. The theory of \(p\)-adic \(L\)-functions is then used to derive Dénes' results more simple since the logarithmic differential quotient may be replaced by the \(p\)-adic logarithm. A basis for the units of the \(p\)-th cyclotomic field is constructed which essentially corresponds to the decomposition of the units mod \(p\)-th powers via idempotents of the group ring of the Galois group. Let \(M = \max v_p(L_p(1;\omega^i))\). It is shown that if \(\varepsilon\) is a unit and \(\varepsilon \equiv 1 \bmod p^{M+1}\), then \(\varepsilon\) is a \(p\)-th power. As an application, it is proved that if \(p\) is irregular and Vandiver's conjecture holds then \(x^p+y^p=z^{p^M}\), \(p\mid z\), \((x,y,z) = 1\), has no non-trivial integral solutions.