Description This subject develops algebraic thought to a high level. The abstract concepts involved in the main topics (group theory and number theory) have many applications in science and technology, and the subject includes an application to cryptography.
School Computer, Data & Math Sciences
Discipline Mathematical Sciences, Not Elsewhere Classified.
Student Contribution Band HECS Band 1 10cp
Check your HECS Band contribution amount via the Fees page.
On successful completion of this subject, students should be able to:
recognise, describe and manipulate the basic structures in abstract algebra and number theory; namely, groups, rings and integral domains
explain the links between these structures and the symmetries of natural objects
apply concepts from group theory and number theory to real life situations, such as RSA cryptography
demonstrate proficiency in both spoken and written mathematical communications, particularly constructing and communicating proofs
1. Number Theory - Divisibility, Euclid's algorithm - Prime numbers - Fundamental Theorem of Arithmetic - Theorems of Fermat and Euler - Congruences - Modular arithmetic - Applications to Cryptography: the RSA cryptosystem
2. Ring Theory - Abstract rings and concrete examples - Integral domains and fields - Rings of polynomials - Polynomial congruences and quotients - Fundamental Theorem of Algebra
3. Group Theory - Abstract groups - Subgroups - Direct products - Isomorphism - Permutations, the Symmetric group - Rubik's cube - Cayley's Theorem - Group of units of a ring - Cosets, Lagrange's Theorem, Quotient groups
Number Theory - Divisibility, Euclid's algorithm - Prime numbers - Fundamental Theorem of Arithmetic - Theorems of Fermat and Euler - Congruences - Modular arithmetic - Applications to Cryptography: the RSA cryptosystem
Ring Theory - Abstract rings and concrete examples - Integral domains and fields - Rings of polynomials - Polynomial congruences and quotients - Fundamental Theorem of Algebra
Group Theory - Abstract groups - Subgroups - Direct products - Isomorphism - Permutations, the Symmetric group - Rubik's cube - Cayley's Theorem - Group of units of a ring - Cosets, Lagrange's Theorem, Quotient groups
Assessment
The following table summarises the standard assessment tasks for this subject. Please note this is a guide only. Assessment tasks are regularly updated, where there is a difference your Learning Guide takes precedence.
Type
Length
Percent
Threshold
Individual/Group Task
Intra-session Exam
1 hour
20
N
Individual
Essay
10 hours
20
N
Individual
Presentation
10 minutes
10
N
Individual
Final Exam
Not specified
50
N
Individual
Prescribed Texts
Nicodemi, Olympia, Sutherland, Melissa A., & Towsley, Gary W. (2007). An introduction to abstract algebra: with notes to the future teacher. Upper Saddle River, N.J: Pearson Prentice Hall.
