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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 K 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 C 2 × C 2 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 C 2 × C 2 fields, J. Number Theory  12 (1980) 160–175]. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new type of relation between knots called the descendant relation. One knot is a descendant of another knot if can be obtained from a minimal crossing diagram of by some number of crossing changes. We explore properties of the descendant relation and study how certain knots are related, paying particular attention to those knots, called fertile knots, that have a large number of descendants. Furthermore, we provide computational data related to various notions of knot fertility and propose several open questions for future exploration. [ABSTRACT FROM AUTHOR]