Module Catalogues, Xi'an Jiaotong-Liverpool University

Module Code: MTH122
Module Title: Introduction to Abstract Algebra
Module Level: Level 1
Module Credits: 5.00
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: 12/10/2019 12:07:09 AM