Gröbner basis for an ideal of a polynomial ring over an algebraic extension over a field and its applications

摘要

An algorithm for a Gröbner basis for an ideal of a polynomial ring over an algebraic extension over a field is presented. Algorithms for inverses of convertible level m(r1,r2,…,rm)-block circulant matrices and minimal polynomials of level m(r1,r2,…,rm)-block circulant matrices over a (s1,s2,…,st)-generated algebra F(α1,α2,…,αt) of degree (n1,n2,…,nt) over a field F are given by the algorithm for the reduced Gröbner basis for an ideal of the polynomial ring F[y1,…,yt,x1,…,xm]. In particular, the inverses of convertible level m(r1,r2,…,rm)-block circulant matrices and the minimal polynomials of level m(r1,r2,…,rm)-block circulant matrices over a (s1,s2,…,st)-generated algebra of degree (n1,n2,…,nt) over the field of all rational numbers can be transformed into those over , and then computed by CoCoA 4.1.