Math 417. Intro to Abstract Algebra
Text: Fraleigh, A First Course in Abstract Algebra is now the default text. Instructors may instead use the free text by Goodman, Algebra: Abstract and Concrete, which can be found here: http://www.math.uiowa.edu/~goodman/algebrabook.dir/algebrabook.html
(a) The Integers Division algorithm. Greatest common divisor. Fundamental theorem of arithmetic. Congruence arithmetic; application to RSA-cryptosystem. 
(b) Permutations Cycle decomposition. Order of a permutation. Even and odd permutations. 
(c) Group Theory Definition and examples. Subgroups, cosets and Lagrange's theorem. Normal subgroups and quotient groups. Homomorphisms. The Isomorphism Theorems. 
(d) Group Actions Cayley's theorem. Burnside's theorem. Conjugacy classes and centralizers. Applications of group actions, eg. to Sylow's theorem or Polya counting. 
(e) Ring Theory I Definition and examples. Polynomial rings. Subrings, ideals and quotient rings. Homomorphisms of rings. The Isomorphism Theorems for rings. Integral domains and fields. Division algorithm for polynomial rings over a field. Roots of polynomials and the Remainder Theorem. The Fundamental Theorem of Algebra (without proof). Maximal ideals in polynomial rings over fields, with application to the construction of fields. 
Exams and leeway = 4 hours.
Total = 43 hours.
Grading: Homework = 25%, exams = 35% , final = 40% .
Prerequisite: either Math 416, or one of Math 347, Math 348, together with one of Math 410, Math 415, CS 273.