Primes of the form p = x2 + ny2 for n ̸≡ 3 mod 4

Specialeforsvar: Anna Mai Østergaard

Titel: Primes of the form p = x2 + ny2 for n ̸≡ 3 mod 4

Abstract: In this thesis, we are going to consider when it is possible to write a prime p on the form p = x2+ny2, with x, y ∈ Z for a fixed natural number n ∈ N. We will prove a concrete description of when this happens in the case of n ̸≡ 3 mod 4. To do this, we will employ algebraic number theory, as well as results from class field theory. In this thesis, we will, however, not prove results form class field theory. The main theorem will be a criterion for the representability of p in the form p = x2 + ny2 in terms of the solvability of a certain polynomial modulo p. After the main theorem it is natural to ask for a concrete decision algorithm for determining the class number. This leads us to consider an equivalence relation on the set of quadratic forms, and we show that the set of equivalence-classes are in bijection with the elements of the class group. Furthermore, we will give an algorithm for finding the number of equivalence-classes, and thus the class number. Lastly, we will apply the theory for a number of concrete n.

Vejleder: Ian Kiming
Censor:   Tom Høholdt, DTU