Power integral bases for Selmer-like number fields

Abstract

The Selmer trinomials are the trinomials over . For these trinomials we show that the ideal has height two and contains the linear polynomial . We then give several necessary and sufficient conditions for to be a regular ring, where is an arbitrary polynomial over a Dedekind domain D such that its ideal C has height two and contains a product of primitive linear polynomials. We next specialize to the Selmer-like trinomials and over D and give several more such necessary and sufficient conditions (among them is that C is a radical ideal). We then specialize to the Selmer trinomials over and give quite a few more such conditions (among them is that the discriminant of is square-free (respectively of is square-free)). Finally, we show that is never square-free when and , but, otherwise, both are very often (but not always) square-free.

Department(s)

Mathematics

Document Type

Article

DOI

https://doi.org/10.1016/j.jnt.2006.01.012

Keywords

content of a polynomial, dedekind domain, discriminant, mathematica program, Noetherian ring, power integral basis, prime ideal, radical ideal, ramify, regular ring, resultant, selmer trinomial

Publication Date

2006

Journal Title

Journal of Number Theory

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