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To give a rigorous introduction to Rings and Fields.
Modular Arithmetic, Euclidean algorithm and Euclid's Lemma.
Concrete Rings: Polynomial rings and Matrix rings. The abstract definition of a ring. Elementary concepts: subrings, ideals, quotient rings. Divisibility in rings. Prime and irreducible elements.
Concrete Fields: the rationals, reals and complex numbers. The integers mod p, p a prime. The abstract definition of a field.
Euclidean Domains, Principal Ideal Domains, Unique Factorization Domains. Gaussian Integers and Applications. Gauss’ Lemma, Irreducibility of Polynomials.