Abstract
Let \(K/\mathbb {F}_q (T)\) be a finite function field extension and denote by O K the integral closure of \(\mathbb {F}_q [T]\) in K. In this article, we are interested in Pólya fields, that is, fields K, such that the O K -module Int(O K ) of integer-valued polynomials over O K admits a regular basis. We show that the cyclotomic extensions of \(\mathbb {F}_q (T)\) are Pólya fields, and we characterize some totally imaginary extensions which are Pólya fields. Then, we are interested in Pólya fields K which have a regular basis of the form \(\left\{\prod_{0\le k < n}\frac{X-a_k}{a_n-a_k},\,n\in\mathbb {N}\right\}\) for some sequences \((a_n)_{n\in\mathbb {N}}\) of elements of O K . For totally imaginary extensions, we show that it is the case if and only if O K is isomorphic to \(\mathbb {F}_q [T]\). This gives a answer to a question raised by Thakur.
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The author thanks his thesis adviser Jean-Luc Chabert, and Mireille Car for their help, and their valuable advices to do this work. The author thanks also the referee for his valuable remarks.