MTH 571 Algebra I (4 credits)Groups, Sylow theorems, solvable and simple groups, free groups, generators and relations of groups, finitely generated abelian groups, semi-direct products, extension of groups, rings, polynomial and power series rings, maximal and prime ideals of commutative rings, Euclidean domains, principal ideal domains, and unique factorization domain. Prerequisite: A course in abstract algebra. MTH 572 Algebra II (4 credits)Injective modules, projective modules, tensor products, modules over PID, decomposition theorems, linear algebra, field extensions, finite fields, geometric constructions, Galois theory, solvability by radicals, computing Galois groups of polynomials. Prerequisite: MTH 571. MTH 575 Elliptic Curves (3 credits)Introduction to elliptic curves, group law, torsion and rank, good and bad reduction, conductor, rational points on elliptic curves, the use of elliptic curves in cryptography. Prerequisite: MTH 572 or permission of instructor. MTH 576 Algebraic Curves (3 credits)Introduction to algebraic curves, coordinate ring, function field, divisors, Jacobian of curves. Moduli spaces of curves, theta functions on curves, automorphisms, etc. Prerequisite: MTH 575 or MTH 572 or permission of instructor. APM 577 Computer Algebra I (4 Credits)A study of the mathematics and algorithms which are used in symbolic algebraic manipulation packages. Topics include computer representation of symbolic mathematics, polynomial ring theory, field theory and algebraic extensions, modular and p-adic methods, subresultant algorithm for polynomial GCD's, Groebner bases for polynomial ideals and Buchberger's algorithm, factorization and zeros of polynomials. Prerequisite: A course in abstract algebra. APM 578 Computer Algebra II (4 Credits)A study of the mathematics and algorithms which are used in symbolic algebraic manipulation packages. Topics include computer representation of symbolic mathematics, polynomial ring theory, field theory and algebraic extensions, modular and p-adic methods, subresultant algorithm for polynomial GCD's, Groebner bases for polynomial ideals and Buchberger's algorithm, factorization and zeros of polynomials. Prerequisite: APM 577. MAT 631 Introduction to Cryptography (3 credits)An introduction to cryptosystems, RSA, discrete log problem, elliptic curve cryptography, hyperelliptic curve cryptography. How safe is safe? New cryptosystems versus old. Prerequisite: APM 673 Coding Theory (4 credits)Linear codes, non-linear codes, B.C.H. codes, dual codes and their weight distribution, perfect codes and cyclic codes. Additional topics drawn from Reed-Solomon codes, Justessen codes, M.D.S. codes, Reed-Muller codes, Golay codes, self-dual codes and invariant theory. Prerequisite: MTH 572 or knowledge of field theory. MTH 670 Algebraic Number Theory (4 credits)Algebraic number fields, integrality and Notherian properties, Dedekeind Domains, Extensions, ramified and non-ramified extensions, ramification in Galois extensions, class groups and units, cyclotomic fields, L-functions, Dedekind zeta-function, Brauer relations. Prerequisite: MTH 572. MTH 671 Commutative Algebra (4 credits)Rings and ideals, modules, exact sequences, tensor products, integral dependence and valuations, the going-up and going-down theorem, chain conditions, Noetherian rings, discrete valuation rings, Dedekind domains. Prerequisite: MTH 572. MTH 672 Algebraic Geometry (4 credits)Algebraic varieties, maps between varieties, Hilbert's Nullstellensatz, Zariski topology, abelian varieties, the Riemann-Roch theorem, Jacobians of curves, sheaves and cohomology, etc. Prerequisite: MTH 671. MTH 695: Special Topics in Cryptography 3MTH 694: Seminar in Mathematics of Communications 2 MAT 699: Thesis Research 3-6 |