# Power integral bases for Selmer-like number fields

#### Abstract

The Selmer trinomials are the trinomials f (X) ∈ {Xn - X - 1, Xn + X + 1 | n > 1 is an integer} over Z. For these trinomials we show that the ideal C = (f (X), f′ (X)) Z [X] has height two and contains the linear polynomial (n - 1) X + n. We then give several necessary and sufficient conditions for D [X] / (f (X) D [X]) to be a regular ring, where f (X) 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 b Xn + c X + d and b Xn + c Xn - 1 + d 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 Z and give quite a few more such conditions (among them is that the discriminant Disc (Xn - X - 1) = ± (nn - (1 - n)n - 1) of Xn - X - 1 is square-free (respectively Disc (Xn + X + 1) = ± (nn + (1 - n)n - 1) of Xn + X + 1 is square-free)). Finally, we show that nn + (1 - n)n - 1 is never square-free when n ≡ 2 (mod 3) and n > 2, but, otherwise, both are very often (but not always) square-free.

*This paper has been withdrawn.*