MATH 235 Lecture : MATH 235 Notes.pdf
19 Nov 2012
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Eyal goren, department of mathematics and statistics, mcgill university. The inverse function: cardinality of a set, number systems. The fundamental theorem of algebra: fields and rings - de nitions and rst examples. Some formal consequences of the axioms: exercises. Arithmetic in z: division, gcd and the euclidean algorithm. The euclidean algorithm: primes and unique factorization. Applications of the fundamental theorem of arithmetic: exercises. Congruences and modular arithmetic: relations, congruence relations. Public key cryptography; the rsa method: exercises. Polynomials and their arithmetic: the ring of polynomials, division with residue, arithmetic in f[x] Some remarks about divisibility in a commutative ring t. Roots of polynomials over z/pz: the first isomorphism theorem. The dihedral group: the theorem of lagrange. Some applications to combinatorics: cauchy"s theorem: a wonderful proof, the rst isomorphism theorem for groups. It is important to realize that algebra started in antiquity as an applied science.
