Cyclotomic units over finite fields (Q1897570)

The paper focusses on determining the size \(S\) of the image of the group of all cyclotomic units in \(\mathbb{Q}(\zeta_p)^+\) after reducing \(\mathbb{Z}[\zeta_p]^+\) modulo a prime \(q\neq p\). Knowing \(S\), some indices which naturally come up when studying units in integral group rings \(\mathbb{Z}[C]\) of cyclic groups \(C\) of order \(pq\) (see e.g. \textit{K. Hoechsmann} [Manuscr. Math. 75, 5--23 (1992; Zbl 0773.16016); Can. J. Math. 47, 113--131 (1995; Zbl 0827.16022)]) become accessible for computation. Having this application in mind, the paper starts out from a cyclic Galois algebra \(E\) over \(\mathbb{F}_q\) with group \(\langle\sigma\rangle\). Let \(\sigma^g\) act on \(E\) as the Frobenius automorphism \(e\mapsto e^q\) and have order \(f\). Then the group \(U\) of units in \(E\) is a free rank-one module for \(R=\mathbb{Z}[x]/\langle q^f-1,x^g-q\rangle\). Each \(v\in U\) gives rise to a defect \(d_q(v)= [U^{(j)}: Rv]\) with \(j=Nv\) and \(U^{(j)}= \{u\in U: Nu^j=1\}\), \(N=\sigma\)-norm. The main result gives the prime divisors of \(d_q(v)\). Of particular interest is a \(v\) such that \(Rv\) is the reduction modulo \(q\) of the \(p\)-th cyclotomic units.