Lucas-Lehmer test

The aim of this thesis is to explain quadratic number field theory and prove correctness of the Lucas-Lehmer primality test. A quadratic number field is a field of the form Q( √ m). Chapter one describes elementary properties of such field's ring of integers focusing on characterisation of the ring's group of units. Chapter two studies ideal factorisation in this ring. It contains proofs of a theorem on unique factorisation of the ideals into prime ideals and a theorem describing all prime ideals. Chapter three employs quadratic number field theory to prove correctness of the Lucas-Lehmer prime test, which is a deterministic primality test for numbers of the form 2p − 1.