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abelian group addition and multiplication algebra assume basis bijection called Clearly closed with respect coefficients cogredient commutative ring Corollary cosets relative deduce defined denoted direct sum division algorithm element of F elementary divisors elementary matrices elementary row transformations equal equivalent Example expressed fi(x field F finite field finite number Hence holds homomorphism ideal idempotents identity element invariant factors inverse invertible matrix irreducible polynomial isomorphic left coset Lemma Let F Let f(x Let G linear combination linearly independent m x n minimal polynomial modulo monic irreducible polynomials n x n matrix nonsingular nonzero element number of elements obtain permutation polynomial of degree polynomials in F[x positive integer primitive idempotents primitive polynomial q elements quadratic form rational number real numbers relative to H relatively prime right coset roots satisfies subfield subgroup of G subset subspace surjective symmetric matrix vector space