Resultants of cyclotomic polynomials (Q5896287)

For \(\zeta\) a primitive \(n\)-th root of unity, it is well known that \(\mathbb{Z}[\zeta]\) is the ring of integers of \(\mathbb{Q}(\zeta)\), and the usual proof uses the canonical decomposition of \(n\) into prime powers. In the note under review, the author presents a new proof which avoids the decomposition of \(n\) into primes by showing that \(\mathbb{Z}[\zeta]\) is a Dedekind ring; he uses the fact that a Noetherian integral domain is a Dedekind ring if for any maximal ideal \(P\) of \(R\), the localization \(R_P\) of \(R\) at \(P\) is a discrete valuation ring. Some facts are proved about \(\mathrm{Res}(\Phi_m,\Phi_n)\), the resultant of two cyclotomic polynomials, and a key result of the author is to show that if \(\Phi_n=\Phi_m^{\phi(p^e)}+p g\) with \(g\in\mathbb{Z}[X]\), \(n=p^e m\), \(p\) a prime not dividing \(m\), then \(g(\zeta)\) is a unit of \(\mathbb{Z}[\zeta]\).