Galois Theory: Lectures Delivered at the University of Notre Dame, Issue 2, Part 1

a₂ abelian group assume augmented matrix axioms B₁ B₂ belong to G C₁ called coefficients column vectors Corollary cosets cyclic group degree denote dependence determinant distinct divisor elements of G exists extension field extension of F f and f F contains F(a₁ F₁ factor group factor of p(x field F field of p(x finite field finite number fixed field fixed point follows function G₁ group G hence homogeneous equations homomorphism integer intermediate field irreducible equation irreducible factors irreducible polynomial isomorphism leave F fixed Lemma linear combination m₁ mod q n₁ non-trivial solution normal extension normal subgroup nth root P₁(x permutation phism polynomial f(x polynomial in F primitive nth root prove repeated roots root of unity roots of p(x satisfied separable polynomial solvable group splitting field subfield subgroup of G suppose symmetric group Theorem 13 u₁ u₂ unique vector space x₁ σ₁