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

Abstract

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( r 1, r 2,…, r m )-block circulant matrices and minimal polynomials of level m( r 1, r 2,…, r m )-block circulant matrices over a ( s 1, s 2,…, s t )-generated algebra F( α 1, α 2,…, α t ) of degree ( n 1, n 2,…, n t ) over a field F are given by the algorithm for the reduced Gröbner basis for an ideal of the polynomial ring F[ y 1,…, y t , x 1,…, x m ]. In particular, the inverses of convertible level m( r 1, r 2,…, r m )-block circulant matrices and the minimal polynomials of level m( r 1, r 2,…, r m )-block circulant matrices over a ( s 1, s 2,…, s t )-generated algebra Q(α 1,α 2,…,α t) of degree ( n 1, n 2,…, n t ) over the field Q of all rational numbers can be transformed into those over Q , and then computed by CoCoA 4.1.