Power integral bases for Selmer-like number fields |
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Authors: | Louis J Ratliff Jr David E Rush |
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Institution: | a Department of Mathematics, University of California, Riverside, CA 92521-0135, USA b Department of Mathematics, Southwest Missouri University, Springfield, MO 65802, USA |
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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))ZX] has height two and contains the linear polynomial (n−1)X+n. We then give several necessary and sufficient conditions for DX]/(f(X)DX]) 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 bXn+cX+d and bXn+cXn−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. |
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Keywords: | primary 12A40 12E10 12F05 13F05 13H05 secondary 12-04 12B10 |
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