Ramification indices, differents and integral bases for radical extensions. II (Q1193260)

This paper is a continuation of the author's previous paper [same journal 56, 569-585 (1991; Zbl 0735.11062)]. Let \(K_ 0\) be the quotient field of a Krull domain \({\mathfrak o}_ 0\). Let \(K/K_ 0\) be an extension of prime degree \(p\) such that \(K=K_ 0(y)\) with \(y^ p=B\in{\mathfrak o}_ 0\) and \({\mathfrak O}\) the integral closure of \({\mathfrak o}_ 0\) in \(K\). If a prime divisor \({\mathfrak p}\) of \(p\) satisfies the condition that \({\mathfrak p}\) is prime to \(B\) and \(B^ p\equiv B\pmod {{\mathfrak p}^{e_{\mathfrak p}+\nu'}}\), where \(e_{\mathfrak p}=v_{\mathfrak p}(p)\) and \(\nu'\) is the least natural number with \(e_{\mathfrak p}+\nu'<\nu'\cdot p\), then \({\mathfrak p}\) is called of \(C\)-type. The author studies the existence of a relative integral basis of \({\mathfrak O}\) over \({\mathfrak o}_ 0\), especially, a \({\mathfrak p}\)-integral basis of \(K/K_ 0\) for a prime \({\mathfrak p}\) of \(C\)- type, and constructs such a basis under some conditions. The proof consists of calculations of ramification indices of prime divisors \({\mathfrak P}\) of \({\mathfrak p}\), \(v_{\mathfrak P}(y-B)\), and the discriminant of \(K/K_ 0\). --- The case where \(K/K_ 0\) is of degree \(p^ d\) is also discussed.