Abstract Algebra

a₁ Abelian group addition and multiplication algebraic automorphism Axiom b₁ binary operations c₁ canonical basis characteristic polynomial column commutative conclude congruence consider contains Corollary correspondence cosets cyclic defined DEFINITION denoted Determine diagonal e₁ elementary matrices elements of F equal equation equivalence relation Example F₁ Find finite function GF(p group G H₁ h₂ Hence identity element implies induction hypothesis integral domain inverse irreducible irreducible polynomial isomorphic linear transformation linearly independent m₁ mathematical matrix modulo monic natural numbers nonsingular nonzero elements normal subgroup obtain one-to-one mapping operation table p₁ pair positive integer Problem proof properties prove quadratic quotient R₁ rational integers rational number reduced row echelon relatively prime ring row echelon form scalar semigroup Show solution splitting field subfield subgroup of G subgroup of order subring subset u₁ unique V₁ V₁(F vector space w₁ zero mod zeros modulo λ² λ³