Abstract: One of the most fundamental theorems in arithmetic is the aptly named Fundamental Theorem of Arithmetic, which asserts that every integer can be uniquely expressed as a product of powers of prime numbers. The same principle holds for the ring of Gaussian integers consisting of integer linear combinations of 1 and i, but often fails for more general quadratic rings obtained by adjoining to the integers the square root of a negative integer. The full classification of such quadratic rings that obey unique factorisation was conjectured by Gauss in his Disquisitiones Arithmeticae but only established in the 20th century by marshalling a remarkable array of sophisticated ideas in the theory of elliptic curves and modular forms. I will discuss these ideas and their extension to a wider class of quadratic rings obtained by adjoining the square root of a positive integer, where the issues become more subtle and many more questions remain open.

Tea will be served at 3:30pm in GAB 461 before the talk.