About the embedding of a number field in a Pólya field
Let \(K\) be a number field, \(O_K\) its ring of integers and \(h(K)\) its class number. Let \(\mathrm{Int}(OK)=\{P\in K[X], P(OK)\subset OK\}\) be the set of integer-valued polynomials on \(O_K\). The field \(K\) is said to be a Pólya field if the \(O_K\)-module \(\mathrm{Int}(OK)\) has a basis \((P_n)_{n\in \mathbb{N}}\) such that for all \(n\), \(P_n\) has degree \(n\). Let \(I_n(K)\) be the subset of \(K\) formed by \(0\) and the leading coefficients of the polynomials of \(\mathrm{Int}(OK)\) having degree \(n\). It is well known that \(I_n(K)\) is a fractional ideal of \(K\) (called the characteristic ideal of index \(n\) in \(K\)), and \(K\) is a Pólya field if and only if all \(I_n(K)\) are principal. Therefore, if \(h(K)=1\), then \(K\) is a Pólya field. Other well-known examples of Pólya fields are cyclotomic fields. A finite extension \(L/K\) is called a Pólya extension if all the extended fractional ideals \(I_n(K)O_L\) are principal in \(L\). Now a classical result of \textit{E. S. Golod} and \textit{I. R. Shafarevich} [Izv. Akad. Nauk SSSR, Ser. Mat. 28, 261--272 (1964; Zbl 0136.02602)] states that the following question has a negative answer: Is every \(K\) contained in a number field \(L\) with \(h(L)=1\) ? (which is equivalent to the finiteness of the Hilbert class field tower). The author of the present paper considers the analogous question: Is every \(K\) contained in a number field \(L\), which is a Pólya field ? The main result gives a positive answer: \(L=H_K\), the Hilbert class field of \(K\); furthermore, the author gives the minimal degree of a Pólya field containing \(K\). The proof uses the notion of Pólya extension (if \(L/K\) is a such extension, \(L\) is analogous to \(H_K\)).
scientific article; zbMATH DE number 6343982