MAT3233

Modern Algebra

MAT 3233 Modern Algebra. (3-0) 3 Credit Hours.

Prerequisites: MAT2233 and MAT3003.

An introduction to modern algebra building up from concrete examples in elementary algebra and number theory which lead to the abstract theory of groups, rings, and fields. Topics include: Arithmetic congruences in the ring of integers; residue rings ℤₙ; finite fields 𝔽ₚ; the group of units 𝑈ₙ; cyclic groups; the Chinese Remainder, Fermat’s and Euler’s theorems; polynomial rings; the Fundamental Theorem of Algebra; irreducible polynomials and factorization in 𝔽[x] and ℤ[𝑥]; quotient rings and construction of the Galois fields 𝐺𝐹(𝑝ⁿ).

Basic examples of groups. Cyclic groups. Permutation groups and cycle decompositions. Group homomorphisms. Normal subgroups, factor groups, and direct products of groups. Definitions and basic examples of rings, integral domains and fields. Ring homomorphisms. Ideals, factor rings, and direct products of rings. Unique factorization of integers and polynomials. Generally offered: Fall, Spring, Summer. Differential Tuition: $150.

Textbook

Lindsay N. Childs, A Concrete Introduction to Higher Algebra (3rd. ed.) Springer-Verlag (2009). ISBN: 978-0-387-74527-5

Week	Chapters	Topics	Student learning outcomes
1	1–2	Integers. Inductive proofs.
2	3–4	Divisibility. The Euclidean Algorithm. Unique factorization of integers.
3	5–6	Congruences and congruence classes. Modular rings ℤₙ.
4	7 & 9	Rings and fields. Theorems of Fermat and Euler.
5	11–12	Groups. The Chinese Remainder Theorem.
6	None	Review. First midterm exam.
7	13–14	Polynomials: division and factorization. Divisibility and the Euclidean Algorithm.
8	15 & 17	The Fundamental Theorem of Algebra. Polynomial congruences. The Chinese Remainder Theorem in 𝔽[𝑥].
9	19-20	Primitive roots and cyclic groups. Carmichael numbers.
10	None.	Review. Second midterm exam.
11	21	Quadratic Reciprocity
12	23	Polynomial congruences and quotients of 𝔽[𝑥].
13	24	Homomorphisms. Finite fields.
14	26	Factorization in ℤ[𝑥]. Irreducible polynomials.