Topics in Algebra

a₁ abelian group algebraic over F automorphism characteristic roots coefficients commutative ring complex numbers conjugate COROLLARY cosets cyclic defined DEFINITION degree division ring divisor element of F equation Euclidean ring example extension of F field F field of rational finite group finite number finite-dimensional given greatest common divisor group G group of order Hermitian homomorphism integral domain invertible irreducible isomorphism kernel Lemma Let G linear transformation linearly independent mapping matrix minimal polynomial multiplication nilpotent normal subgroup number of elements one-to-one orthonormal permutation polynomial p(x positive integer prime number Problem Proof prove rational numbers real numbers relatively prime result right cosets satisfies solvable space over F splitting field subfield subgroup of G subset subspace Suppose symmetric Theorem unit element V₁ vector space w₁ whence απ