MATH30130

Learning Outcomes:

The student expected develop an understanding of the following concepts:

(i) Basic structures in Algebraic Number Theory (Euclidean Domains, PIDs, UFDs, Fundamental Theorem of Arithmetic for certain rings)

(ii) Fermat's method of infinite descent

(iii) Proof of Fermat's Last Theorem for small exponents based on field arithmetic

(iv) Gauss' reciprocity law and classification of conics over finite fields

(v) Point counting over finite fields (Chevalley-Warning Theorem).

(vi) p-adic numbers, basic p-adic analysis, multiplicative structure of p-adic fields, Hensel's Lemma, conics over p-adic fields and Hilbert symbol.

(vii) Hasse-Minkowski theorem and classification of conics over number fields.

Indicative Module Content:

Basic structures in Algebraic Number Theory (Euclidean Domains, PIDs, UFDs, Fundamental Theorem of Arithmetic for certain rings), Gauss' reciprocity law, Chevalley-Warning Theorem, p-adic numbers, basic p-adic analysis, multiplicative structure of p-adic fields, Hensel's Lemma, conics over p-adic fields and Hilbert symbol, Hasse-Minkowski theorem and classification of conics over number fields.

Student Effort Hours:

Approaches to Teaching and Learning:

Requirements, Exclusions and Recommendations

Contents of MATH10040 Numbers and Functions (mathematical induction; solutions of inequalities; basic concepts associated with function; greatest common divisors; theorems of Fermat and Euler; basic concepts of Number Theory, such as prime numbers and their relevance to codes)

Contents of MATH20310 Groups, Rings and Fields (definition and examples of groups, subgroups, cosets and Lagrange's Theorem, the order of an element of a group, normal subgroups and quotient groups, group homomorphisms and the isomorphism theorems, definitions of a commutative ring with unity, integral domains and fields, units, irreducibles and primes in a ring, ideals and quotient rings, prime and maximal ideals, ring homomorphisms and the homomorphism theorem, polynomial rings, the division algorithm, gcd for polynomials, irreducible polynomials and field extensions)

Module Requisites and Incompatibles

Assessment Strategy

Feedback Strategy/Strategies

• Feedback individually to students, post-assessment
• Group/class feedback, post-assessment
• Self-assessment activities

How will my Feedback be Delivered?

Self-assessment activities are tentative and will be based on the following model: Students will be asked to grade their own responses to select assignments, based on a detailed solution sheet. They will prepare a detailed report on their weaknesses on key aspects of the module. Based on the accuracy of this report, their homework grades will be positively updated. The other two feedback strategies to employed (Feedback individually to students, post-assessment and Group/class feedback, post-assessment) are self-explanatory.