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Department of Mathematical Sciences

# MATH3031 Number Theory III

With the resolution of Fermat's Last Theorem by Andrew Wiles, the study of Diophantine Equations (equations where the unknowns are to be integers) has come again to the forefront of mathematics. In this course we study some of the techniques, mainly based on ideas of factorization and congruence, which have been applied to the Fermat and other Diophantine equations. We find, for example, the number of solutions in positive integers to

x³+y³=z³ (the case n=3 of the Fermat equation)
x²=y³-2 (a case of Mordell's equation)

(The answers are 0 and 1 respectively.)

Our methods lead us to study number systems (rings) other than that of the ordinary integers (e.g. the Gaussian integers), and we consider what we can do when things do not go according to plan, that is, when unique factorization fails. E.g. recall that in the ring ℤ[√-5] we have 6=2×3=(1+√-5)×(1-√-5). When uniqueness of factorization breaks down in this way, we are led to consider ideals as "ideal numbers" (this is the reason why ideals are called ideals). We use this idea to regain a form of uniqueness of factorization.

We introduce the ideal class group which measures by how much unique factorization fails in number rings. Using a pretty application of the "Geometry of Numbers", we show that there are only finitely many ideal classes. This leads to the solution of more sorts of diophantine equations. We also study the units of these number rings. These are given by the Dirichlet unit theorem which can be viewed as a generalization of the characterization of the solutions to Pell's equation x²-Dy²=1.

We finally turn to some analytic tools and study the zeta function of quadratic fields, which generalize the famous Riemann zeta function. In particular, we discuss the (analytic) class number formula.

## Outline of Course

Aim: To provide an introduction to Algebraic Number Theory (Diophantine Equations and Ideal Theory).

### Term 1

• Quick review of ring theory: Ideals (prime and maximal ideals). Factorization in monoids (prime vs. irreducible). Ring is Euclidean ⇒ PID ⇒ UFD. Preliminary discussion of ideal factorization.
• Fields: Algebraic number fields. Vector spaces over such. Field extensions: index; tower theorem; simple extensions and minimum polynomials; Conjugates over ℚ. Homomorphisms of fields. Norms and traces over ℚ. Matrix formulae for these.
• Algebraic Integers: The several characterizations of algebraic integers OK in a number field K; OK is a ring. Integrality of norm and trace.
• The Discriminant and Integral Bases: Definition, formal properties and calculation of the discriminant of a basis of a field over ℚ.
• Factorization of Ideals: Fractional ideals. Unique factorization and inverses of ideals. The ideal group. Prime ideals.
• Quadratic Fields and Integers: Easy Euclidean quadratic fields. Representation of integers by binary quadratic forms where there is unique factorization in the relevant quadratic field.

### Term 2

• The Ideal Class Group: The class group. Minkowski's theorem. Finiteness and determination of the class group. Representation of integers by definite binary quadratic forms (problems rather than general theory).
• Dirichlet's Unit Theorem. Representation of integers by non-definite binary quadratic forms.
• L-functions: Riemann zeta function, Dirichlet L-series, zeta function for quadratic fields. Basic properties and analytic continuation.
• Class number formula for quadratic fields.

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.