Ramanathan K.G. Lectures on the Algebraic Theory of Fields

India.: Bombay. Tata Institute of Fundamental Research, 1956. — 222 p., eBook, English (Interactive menu).There are notes of course of lectures on Field theory aimed at providing the beginner with an introduction to algebraic extensions, algebraic function fields, formally real fields and valuated fields. These lectures were preceded by an elementary course on group theory, vector spaces and ideal theory of rings—especially of Noetherian rings. A knowledge of these is presupposed in these notes. In addition, we assume a familiarity with the elementary topology of topological groups and of the real and complex number fields.
Most of the material of these notes is to be found in the notes of Artin and the books of Artin, Bourbaki, Pickert and Van-der-Waerden.
My thanks are due to Mr. S. Raghavan for his help in the writing of these notes.General extension fields.
Extensions.
Adjunctions.
Algebraic extensions.
Algebraic Closure.
Transcendental extensions.
Algebraic extension fields.
Conjugate elements.
Normal extensions.
Isomorphisms of fields.
Separability.
Perfect fields.
Simple extensions.
Galois extensions.
Finite fields.
Algebraic function fields.
F.K. Schmidt’s theorem.
Derivations.
Rational function fields.
Norm and Trace.
Norm and trace.
Discriminant.
Composite extensions.
Kronecker product of Vector spaces.
Composite fields.
Applications.
Special algebraic extensions.
Roots of unity.
Cyclotomic extensions.
Cohomology.
Cyclic extensions.
Artin-Schreier theorem.
Kummer extensions.
Abelian extensions of exponent p.
Solvable extensions.
Formally real fields.
Ordered rings.
Extensions of orders.
Real closed fields.
Completion under an order.
Archimedian ordered fields.
Valuated fields.
Valuations.
Classification of valuations.
Examples.
Complete fields.
Extension of the valuation of a complete.
Fields complete under archimedian valuations.
Extension of valuation of an incomplete field.
Appendix.
Decomposition theorem.
Characters and duality.
Pairing of two groups.