MAT3233
Modern Algebra
MAT 3233 Modern Algebra. (30) 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 ℤ 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).
Textbook
Lindsay N. Childs, A Concrete Introduction to Higher Algebra (3rd. ed.) SpringerVerlag (2009). ISBN: 9780387745275
Topics
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 
 
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 
1920 
Primitive roots and cyclic groups. Carmichael numbers. 

10 
 
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. 

15 
 
Wrapup and review. Student Study Day. 