Rédei: Algebra

 Set theory (nonaxiomatic); several forms of Zorn's lemma.
 The concept of an algebraic structure (a set equipped with some operations, usually binary), semigroups, rings, skew fields, homomorphism, quotient with respect to an equivalence relation, the JordanHölderSchreier theorem.
 Structures with operators, universal algebraic considerations, free structures, polynomial rings, determinants, quaternions.
 Euclidean rings, ideals, divisibility, Euclidean algorithm.
 Finite abelian groups, the fundamental theorem; Hajos' theorem.
 Modules, vector spaces, matrices, elementary divisors, finitely generated abelian groups.
 Rings of polynomials, zero divisors, derivatives, multiple factors, symmetric polynomials, interpolation, Eisenstein's theorem, ideals in commutative rings.
 Field theory, extensions, normality, cyclotomy, finite fields, Wedderburn's theorem, transcendental extensions, separable extensions, norm and trace.
 Ordered structures, archimedean order, absolute value.
 Fields with valuations, real numbers, real closed fields, nonarchimedean valuations, Ostrowski's theorem, the Hensel lemma.
 Galois theory, quadratic reciprocity, cyclic fields, solvability, the general equation, solution of cubic and quartic equations, geometric constructions, the normal basis theorem.
JOC/EFR August 2007
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