Pólya fields, Pólya groups and Pólya extensions: a question of capitulation
For an algebraic number field \(K\) denote by \(\text{Int}(K)\) the set of all integral-valued \(K\)-polynomials, i.e., \(\text{Int}(K)=\{f\in K[X]:\;f(Z_K)\subset Z_K\}\), \(Z_K\) denoting the ring of integers of \(K\). Moreover let \(I_n(K)\) be the fractional ideal generated by the leading coefficients of \(f\in \text{Int}(K)\) with \(\deg f=n\). A field \(K\) is called a Pólya field, if the \(Z_K\)-module \(\text{Int}(K)\) has a basis \(\{f_n\}\) with \(\deg f_n=n\). \textit{G. Pólya} [J. Reine Angew. Math. 149, 97--116 (1919; JFM 47.0163.04)] proved that \(K\) is a Pólya field if and only if all ideals \(I_n(K)\) are principal, and \textit{A. Ostrowski} [J. Reine Angew. Math. 149, 117--124 (1919; JFM 47.0163.05)] showed that this holds if and only if for every prime power \(q\) the product of all ideals with norm \(q\) is principal. The author calls an extension \(L/K\) a Pólya extension, if every ideal \(I_n(K)\) becomes principal in \(L\), and shows that this happens if and only if the \(Z_L\)-module \(\text{Int}(K,L)\) consisting of all polynomials \(f\in L[X]\) satisfying \(f(Z_K)\subset Z_L\) has a basis \(\{g_n\}\) with \(\deg g_n = n\). (Note that in a paper of \textit{M. Spickermann} [Integer valued polynomials and Galois invariant ideals over algebraic number fields. Fachbereich Mathematik der Wilhelms-Universität Münster (1986; Zbl 0628.12002)] another notion of a Pólya field has been considered.)NEWLINENEWLINEThe paper also contains some remarks about the Pólya group \(\text{Po}(K)\) of a field \(K\), defined as the subgroup of the class-group of \(K\) generated by classes of the ideals \(I_n(K)\), and deals with the question when the composite of two Pólya fields is a Pólya field.
scientific article; zbMATH DE number 6075041