Algebra: Volume 4 - Field Theory

Starting with the basic notions and results in algebraic extensions, the authors give an exposition of the work of Galois on the solubility of equations by radicals, including Kummer and Artin-Schreier extensions followed by a Chapter on algebras which contains, among other things, norms and traces of algebra elements for their actions on modules, representations and their characters, and derivations in commutative algebras. The last Chapter deals with transcendence and includes Luroth's theorem, Noether's normalization lemma, Hilbert's Nullstellensatz, heights and depths of prime ideals in finitely generated overdomains of fields, separability and its connections with derivations.