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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