# Rédei: *Algebra*

In 1954

**László Rédei**published Volume 1 of his*Algebra*book in Hungarian. It was published by*Akadémiai Kiadó, Budapest.*Paul Halmos gave the following translation of the chapter headings into English- 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.

Last Updated August 2007