zbMATH — the first resource for mathematics

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.

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
Full Text: DOI EuDML
[1] M. Bhargava, \(P\)-orderings and polynomial functions on arbitrary subsets of Dedekind rings. J. Reine Angew. Math. 490 (1997), 101-127. · Zbl 0899.13022
[2] M. Bhargava, Generalized factorials and fixed divisors over subsets of a Dedekind domain. J. Number Theory 72 (1998), 67-75. · Zbl 0931.13004
[3] H. Bass, Big projective modules are free. Illinois J. Math. 7 (1963), 24-31. · Zbl 0115.26003
[4] N. Bourbaki, Algèbre, Chapitre V. Masson, Paris, 1981.
[5] P.J. Cahen, J.L. Chabert, Integer-valued polynomials. Mathematical Surveys and Monographs 48, Amer. Math. Soc., Providence, 1997. · Zbl 0884.13010
[6] J.L Chabert, Factorial Groups and Pólya groups in Galoisian Extension of \(\mathbb{Q}\). Proceedings of the fourth international conference on commutative ring theory and applications (2002), 77-86. · Zbl 1107.13303
[7] H. Cohen, A Course in Computational Algebraic Number Theory. Springer, 2000. · Zbl 0786.11071
[8] D. Hilbert, Die Theorie der algebraischen Zahlkörper. Jahresbericht der Deutschen Mathematiker-Vereinigung 4 (1894-95), 175-546.
[9] E.S. Golod, I.R. Shafarevich, On the class field tower. Izv. Akad. Nauk, 28 (1964). · Zbl 0136.02602
[10] H. Koch, Number Theory. Graduate Studies in Mathematics, A.M.S., 2000. · Zbl 0953.11001
[11] A. Ostrowski, Über ganzwertige Polynome in algebraischen Zahlkörpren. J. reine angew. Math. 149 (1919), 117-124. · JFM 47.0163.05
[12] G. Pólya, Über ganzwertige Polynome in algebraischen Zahlkörpern. J. Reine Angew. Math. 149 (1919), 97-116.
[13] P. Ribenboim, Classical Theory of Algebraic Numbers. Springer, 2000. · Zbl 1082.11065
[14] J.P Serre, Corps Locaux. Hermann, Paris, 1962.
[15] H. Zantema, Integer valued polynomials over a number field. Manusc. Math. 40 (1982), 155-203. · Zbl 0505.12003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.