Pólya fields and Kuroda/Kubota unit formula

Abstract

Let K be a number field. The Pólya field concept is used to know when the module of integer-valued polynomials over the ring of integers [Formula: see text] of K has a regular basis. In [C. W.-W. Tougma, Some questions on biquadratic Pólya fields, J. Number Theory 229 (2021) 386–398], the author used cohomological results from [C. B. Setzer, Units over totally real [Formula: see text] fields, J. Number Theory 12 (1980) 160–175] to answer questions raised in [A. Leriche, Pólya fields, Pólya groups and Pólya extensions: a question of capitulation, J. Théor. Nr. Bordx. 23 (2011) 235–249; A. Leriche, Cubic, quartic and sextic Pólya fields, J. Number Theory 133 (2013) 59–71] on biquadratic Pólya fields. Here we first prove that number fields were omitted from the list of exceptional fields cited in [A. Leriche, Pólya fields, Pólya groups and Pólya extensions: a question of capitulation, J. Théor. Nr. Bordx. 23 (2011) 235–249; A. Leriche, Cubic, quartic and sextic Pólya fields, J. Number Theory 133 (2013) 59–71]. We therefore identify new biquadratic Pólya fields, where the prime number 2 is totally ramified. This result corrects and completes some others on the literature. On the other hand, we show that the main results of [C. W.-W. Tougma, Some questions on biquadratic Pólya fields, J. Number Theory 229 (2021) 386–398] and this paper can be proved with a single method using Kuroda/Kubota’s unit formula without cohomological results of [C. B. Setzer, Units over totally real [Formula: see text] fields, J. Number Theory 12 (1980) 160–175].

Keywords

• Regular basis
• principal ideals
• ramification
• group of units
• norm equation