Factoring primes and sums of two squares

Abstract

Mathematicians began to study a series of properties about numbers a long time ago, and a new field of mathematics, the number theory, was born from this. Some special properties of numbers in the number theory make mathematicians use the knowledge of group theory to make some ingenious answers when considering some problems. In the analytic number theory, equations related to numbers have always been a concern of mathematicians. The most famous Fermat's last theorem also brought long-term troubles to countless mathematicians and was finally proved by the British mathematician Wiles. Many famous theorems also prove that some problems in the number theory can be solved by thinking in relation to other algebraic knowledge. This paper focuses on the factoring primes and constructs prime ideals of lying above a prim from irreducible factors of . The paper also shows that these are all prime ideals lying above . Based on these theorems and definitions, as a simple application of the theory, this paper first considers which primes can be written as sums of two squares, then the second part of this paper gives the answer: is a sum of two squares if and only if .