Stating 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 Lüroth’s theorem, Nöether’s normalization lemma, Hilbert’s Nullstellensatz, heights and depths of prime ideals in finitely generated overdomains of fields, separability and its connections with derivations.

Chapter 1. Algebraic Extensions: Extensions and algebras / Algebraic extensions / Splitting fields, algebraic closure and normal extensions / Roots of unity, finite fields / Separable and inseparable algebraic extensions / Chapter 2. Galois Theory: The Galois group and the fundamental theorem / Solubility of equations by radicals / Additional results / Chapter 3. Algebras: Algebras / Tensor product of algebras / Norms and traces / Generalities on representations and their characters / Derivations in commutative algebras / Chapter 4. Further Field Theory : Transcendence / Finitely generated overdomains of fields / Separability / Derivations and separability / Bibliography / Index

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