Algebra

abelian group absolute value algebraic extension algebraically closed algebraically independent assume automorphism bilinear form called character characteristic closure coefficients commutative ring complex numbers contained COROLLARY coset cyclic decomposition define degree denote dimension direct sum embedding endomorphism equal exists an element factor finite dimensional finite extension finite number follows free module function functor Furthermore Galois extension Galois group given Hence hermitian homomorphism induced injective integer inverse irreducible polynomial isomorphism kernel left ideals lemma Let f Let G linear map linearly independent matrix maximal ideal monoid morphism multiplicative n-th root Noetherian non-degenerate non-singular non-zero notation prime ideal prime number Proof PROPOSITION prove quotient field R-module representation residue class ring-homomorphism root of unity separable sequence Show subfield subgroup of G submodule subring subset subspace tensor product theorem trivial unit element valuation ring variables vector space write zero