## Math 417. Intro to Abstract Algebra

Instructor Syllabus

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

### Topics covered

(a) The Integers Division algorithm. Greatest common divisor. Fundamental theorem of arithmetic. Congruence arithmetic; application to RSA-cryptosystem. [4]

(b) Permutations Cycle decomposition. Order of a permutation. Even and odd permutations. [3]

(c) Group Theory Definition and examples. Subgroups, cosets and Lagrange's theorem. Normal subgroups and quotient groups. Homomorphisms. The Isomorphism Theorems. [10]

(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. [10]

(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. [12]

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.