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Pólya fields, Pólya groups and Pólya extensions: a question of capitulation. (English. French summary) Zbl 1282.13040
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 \(\deg f_n=n\). 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 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 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.)
The 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.

MSC:
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
11R04 Algebraic numbers; rings of algebraic integers
11R09 Polynomials (irreducibility, etc.)
11R32 Galois theory
11R37 Class field theory
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