Module Catalogues, Xi'an Jiaotong-Liverpool University   
 
Module Code: MTH122
Module Title: Introduction to Abstract Algebra
Module Level: Level 1
Module Credits: 5.00
Academic Year: 2019/20
Semester: SEM2
Originating Department: Mathematical Sciences
Pre-requisites: MTH107
   
Aims
• To introduce some commonly used mathematical structures.


• To provide knowledge about basic results in arithmetic and group theory.


• To help with the transition from concrete to abstract mathematical thinking.


• To present applications, such as data encryption, that rely heavily on abstract mathematics.


• To demonstrate how algebraic structures can be used to unify diverse mathematical topics.


• To develop the students’ skills in reading and writing mathematical proofs.


• To enrich the mathematical vocabulary of the students.

Learning outcomes 
A Apply the Euclidean Algorithm to integers or polynomials.

B Work with fundamental mathematical concepts, such as relations and permutations.

C Use modular arithmetic to solve problems related to cryptography and coding theory.

D Recognise common algebraic structures, such as groups, rings or fields.

E Find the image and kernel of a homomorphism and show (in simple cases) when two structures are isomorphic.

F Apply basic results in group theory, such as Lagrange’s Theorem.

Method of teaching and learning 
This module will be delivered by a combination of formal lectures and tutorials.
Syllabus 
1. Brief review of sets and functions, including inverse function vs. inverse image.

2. Equivalence Relations: Equivalence classes and quotient sets.

3. Arithmetic of Numbers and Polynomials: Primes and the Unique Factorization Theorem. Roots and linear factors of polynomials. Greatest common divisor, Euclid's algorithm and Bezout's lemma.

4. Modular Arithmetic: Solving linear congruencies (modular multiplicative inverses), Chinese remainder theorem, Fermat’s and Euler's Theorem.

5. Public Key Cryptography and Error-detecting/correcting Codes: Principles and examples.

6. Algebraic Structures - Rings and Fields: Basic definitions, properties and examples of rings of matrices and polynomials, residue class rings (Z/nZ), some finite fields. Substructures, homomorphisms and the first isomorphism theorem.

7. Basic concepts of Group Theory: Symmetries of geometric figures. Permutations (cycle notation, order and sign). Lagrange's Theorem. Simple examples of group actions.

8. Research-led examples /questions/conjectures (if time).


Delivery Hours  
Lectures Seminars Tutorials Lab/Prcaticals Fieldwork / Placement Other(Private study) Total
Hours/Semester 39    13      98  150 

Assessment

Sequence Method % of Final Mark
1 Group Work 10.00
2 Midterm Exam 10.00
3 Final Exam 80.00

Module Catalogue generated from SITS CUT-OFF: 8/20/2019 6:23:57 PM