Power integral bases for Selmer-like number fields

JOURNAL OF NUMBER THEORY

卷:121

Article

Ratliff, Louis J., Jr. ; Rush, David E. ; Shah, Kishor

关键词: content of a polynomial;    Dedekind domain;    discriminant;    mathematica program;    Noetherian ring;    power integral basis;    prime ideal;    radical ideal;    ramify;    regular ring;    resultant;    Selmer trinomial;   

DOI  :  10.1016/j.jnt.2006.01.012

来源: Elsevier

【 摘 要 】

The Selmer trinomials are the trinomials f (X) is an element of {X-n - X - 1, X-n + 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 bX(n) + cX + d and bX(n) + cX(n-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(X-n - X - 1) = +/-(n(n) - (1-n)(n-1)) of X-n - X - 1 is square-free (respectively Disc(X-n + X + 1) = +/-(n(n) + (1-n)(n-1)) of X-n +X + 1 is square-free)). Finally, we show that n(n) + (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. (C) 2006 Elsevier Inc. All rights reserved.

【 授权许可】

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