## 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

### Citations:

Zbl 0628.12002; JFM 47.0163.04; JFM 47.0163.05
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### References:

 [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
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