Minimal conductors of Kummer extensions by roots of unit elements (Q2764902)

Let \(p\) be an odd prime number and \(F\) be a finite extension of \(\mathbb{Q}_p\), the field of \(p\)-adic numbers. Assume that the \(p^n\)-th roots of unity \(\mu_{p^n}\) are contained in \(F\). Denote by \(U_F^{(m)}\) the \(m\)-th group of higher principal units, \(m \geq 1\). The author defines the Swan conductor of a non-zero element \(\alpha\) of \(F\) to be the smallest nonnegative integer \(m\), such that the \(p^n\)-th norm residue symbol \((\alpha, \beta)\) is trivial for all \(\beta \in U_F^{(m+1)}\) (this sometimes differs by 1 from the usual notion of conductor). The aim of this paper is to introduce a new approach for computing this conductor. NEWLINENEWLINENEWLINEFor an integer \(i \geq 1\), let \(s_i\) denote the minimum of the conductors \(s(\alpha)\), when \(\alpha \in U_F^{(i)}-U_F^{(i+1)}\) and let \(a_i := (n-v_p(i))e+e/(p-1)-i\) with \(v_p\) the \(p\)-adic valuation of \(\mathbb{Q}_p\) and \(e\) the absolute ramification index of \(F\). With these notations, the author proves (using S. Sen's explicit formulae for the \(p^n\)-th norm residue symbols) that \(s_i = a_i\), if \(a_i \geq 0\) and \(v_p(i) \leq n-1\), and \(s_i = 0\) otherwise. The key step to prove this equality is the following: for any unit \(x\) in \(F\), there exists \(\delta \in U_F^{(i)}-U_F^{(i+1)}\) with \(s(\delta)=s_i\) such that \(\delta \equiv 1+x\pi^i \bmod \pi^{i+1}\). Here \(\pi\) is a fixed prime element of \(F\). Then the author explains how to compute the conductor of an element from this equality. The special case of the cyclotomic fields \(F= \mathbb{Q}_p(\mu_{p^n})\) (with \(p \geq 5\)) is treated in detail. The methods developed in this paper allow to determine the conductors of nonzero elements of \(\mathbb{Q}_p(\mu_{p^n})\) when \(p \geq 5\) and \(n \leq 3\).