A note on ideal bases (Q1823274)

Okutsu obtained integral bases of finite separable extensions of quotient fields of Dedekind domains in terms of divisor polynomials. The author of this paper generalizes this result to ideal bases. Let D be a Dedekind domain with the quotient field k and f(x) a monic irreducible separable polynomial of degree \( n\) in D[x]. Let \(\theta\) be one of the roots of f(x) in the algebraic closure \(\bar k\) of k. Let \(K=k(\theta)\) and assume that A is a fixed nonzero ideal of K. Let \(\{P_ u| u\in U\}\) be the set of all prime ideals of D, \(k_ u\) a completion of k with respect to \(P_ u\) and \(D_ u\) its valuation ring. Assume k is a subfield of \(k_ u\). Let \(\bar k_ u\) be an algebraic closure of \(k_ u\). Denote by \(e_ u( )\) the exponential valuation of \(\bar k_ u\) which is the unique extension of the \(P_ u\)-adic valuation of k. For each \(u\in U\), let \(f(x)=\prod^{s_ u}_{i=1}f_{u,i}(x) \) be a factorization of f(x) in \(k_ u[x]\), where \(f_{u,i}\) is a monic irreducible polynomial in \(D_ u[x]\). Let \(\theta_{u,i}\) be one of the roots of \(f_{u,i}(x)\) in \(k_ u\). Define a k-isomorphism \(h_{u,i}\) from \(K=k(\theta)\) into \(\bar k_ u\) by \(h_{u,i}(\theta)=\theta_{u,i}\). Let \(h_{u,i}(A)=AD_{k_ u(\theta_{u,i})}\) which is an ideal of \(k_ u(\theta_{u,i})\) where \(D_{k_ u(\theta_{u,i})}\) is the valuation ring of \(k_ u(\theta_{u,k})\). For each \(u\in U\), define a rational-valued and \(\infty -valued\) function \(\Phi_ u( )\) on K as follows: \(\Phi_ u(b)=\min \{e_ u(h_{u,i}(b)-e_ u(h_{u,i}(A))\} \), \(b\in K\). Put \(w_{u,m}=\Phi_ u(g_{u,m}(\theta))\) and \(v_{u,m}=[w_{u,m}]\) where [ ] denotes the greatest integer function. Let \(u_ 0=\{u\in U| \quad P_ u\quad is\quad a\quad prime\quad divisor\quad of\quad d(f)\quad or\quad N_{K| k}A\}\) where \(d(f)=(- 1)^{n(n-1)/2}N_{K| k}f'(\theta))\) and \(f'\) denotes the derivative of f(x). For \(u\in U\), let \(c_ u=\max_{1\leq i\leq s_ u}e_ u(h_{u,i}(A)).\) The author proves that for any positive integer \(m<n\), if \(g_ m(x)\) is a monic polynomial of degree \( m\) in D[x] satisfying \(g_ m(x)=g_{u,m}(x)\quad mod\quad P_ u[w_{u,m}+c_ u]+1\) for any \(u\in U_ 0\), then \(A=B_ 0\oplus \oplus^{n-1}_{m=1}B_ m^{-1}g_ m(\theta)v \) where \(B_ 0=A\cap k\) and \(B_ m=\prod_{u\in U}P_ u^{v_{u,m}} \).