Primes That Factor

During grade school, many math students encounter the concept of "prime numbers", integers whose only divisors are one and themselves. In elementary number theory,  working with such classical primes gives us many important theorems and properties. However, we will be looking at the issues that occur when we adjoin the imaginary number i to the integers (i.e., looking at terms that can be expressed in the form a+bi), a space known as the Gaussian integers. Examples of these numbers would include any integer, as well as terms such as 1+i, 2+3i, or 6i. So, what would occur if we examined a prime in the context of the Gaussian integers? For instance, consider the integer 5. Ordinarily, we would say that 5 is a prime number. However, in the Gaussian integers, 5 = (2+i)(2-i). Suddenly, we have that a prime number is actually factorable. But how can we tell which primes split in this manner? We will discuss the tri-fold relationship between a prime splitting, the prime's value modulo 4, and whether the prime can be written as the sum of squares. Then, we will easily be able to classify which primes split in the Gaussian integers. In addition, we will briefly touch upon how this method (albeit along with some extra work) can allow us to make the same deductions about primes in other quadratic extensions.