Abstract
Let A be a semisimple n-dimensional commutative algebra over a field F . It is easy to see that, given a basis B of A , the transposes of the matrices over F that represent a∈ A regularly with respect to B can be simultaneously diagonalized over many fields. Using the multiplication table of the algebra we construct an ideal I of F[x 1,…,x n] given in terms of a Gröbner basis of the ideal I with respect to a total degree lexicographic monomial ordering and show that A is isomorphic to F[x 1,…,x n]/ I . We will then use Gröbner basis properties to prove the properties of the algebra.
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