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Department of Mathematical Sciences

# Seminar Archives

## Gandalf (Pure Maths Student Seminar): Primes of the form x^2 + ny^2

Presented by Steven Charlton, Durham

21 October 2015 16:00 in CM221

Fermat observed that (except for p = 2) a prime p can be written as the sum of two squares if aond only if p = 1 (mod 4). This result motivates our basic question: which primes does a given quadratic form represent?

To begin to answer this, we will relate the question of primes represented by a quadratic form to questions about ideal classes in quadratic number fields. And we will then be able to study these questions using the powerful tools of class field theory.

The main goal of this talk will to give a complete answer to this question for a specific class of quadratic forms, the so-called principal forms x^2 + ny^2. In this case the answer has the following form: there exists a polynomial f_n(t) such that p = x^2 + ny^2 if and only if f_n(t) has a root modulo p. And for squarefree n, this polynomial f_n(t) has an explicit interpretation as the polynomial describing the `Hilbert class field' of Q(sqrt(n)).