Power integral bases for Selmer-like number fields
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.
content of a polynomial, dedekind domain, discriminant, mathematica program, Noetherian ring, power integral basis, prime ideal, radical ideal, ramify, regular ring, resultant, selmer trinomial
Ratliff Jr, Louis J., David E. Rush, and Kishor Shah. "Power integral bases for Selmer-like number fields." Journal of Number Theory 121, no. 1 (2006): 90-113.
Journal of Number Theory