MAT3233

From Department of Mathematics at UTSA
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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: 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

Cyclic groups and cryptography

Example

Chapter 13

The Structure of Groups

Example

Example

Example

Chapter 16

Rings

Example

Example

Example

Chapter 17

Polynomials

Example

Example

Example

Chapter 18

Integral Domains

Example

Example