- Set theory (non-axiomatic); 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 Jordan-Hölder-Schreier 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, non-archimedean 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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