Integral basis of the field $Q\left( {\sqrt[n]{a}} \right)$
Kōsaku Okutsu · Proceedings of the Japan Academy Series A Mathematical Sciences · 1982
In application of the theory exposed in our preceding notes [1],we give here explicitly an integral basis of the field Q(/-ff), where n, a e Z, n_> 2, (n, a)= 1.Some facts about the Newton diagram which are needed, will be first explained.1. Newton diagram and irreducible factors of a polynomial.Let k be complete discrete vluation field with exponential valuation v.For a monic polynomial f(x) x n+ ax+.+ a in k[x], we define the Newton diagram of f(x) as follows (cf.[2]).Put m=v(a) (i=l, .., n).We define inductively a sequence (i0, i, ..., i) which is sub- set of {0, 1, ..., n}, and a sequence of rational numbers (,..., t) as follows.Put i0-0.Assuming i is already defined, we put