Syllabus  MA416/MA491
MA416 Ring Theory

Introductory examples of rings and fields

Axioms. Subrings. Integral domains; theorems of Fermat and Euler

Field of quotients of an integral domain

Division rings. Quaternions.

Rings of polynomials. Factorisation. Gauss's Lemma. Eisenstein's criterion

Ideals, factor rings, ring homomorphisms. Homomorphism theorems

Prime ideals, maximal ideals. Principal ideal rings

Unique factorsation domains, Euclidean domains.

Gaussian integers
MA491 Field Theory

Field extensions  simple, algebraic, transcendental

The degree of an extension

Ruler and compass constructions

Algebraically closed fields, splitting fields and finite fields

Galois groups and the Galois correspondence

Introduction to codes (ISBN, linear, cyclic)
Texts

J.B. Fraleigh, "A First Course in Abstract Algebra" (Addison Wesley)
References

I.T. Adamson, "Introduction to Field Theory" (Oliver & Boyd)

I.N. Herstein, "Topics in Algebra" (Wiley)

I.N. Stewart, "Galois Theory" (Chapman & Hall)
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