On successful completion of this subject, students should be able to:

1. recognise, describe and manipulate the basic structures in abstract algebra and number theory; namely, groups, rings and integral domains
2. explain the links between these structures and the symmetries of natural objects
3. apply concepts from group theory and number theory to real life situations, such as RSA cryptography
4. 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