note that because , the ideals generated by and are the same. A complete answer to the question of which ideals remain prime in the Gaussian integers is provided by Fermat's theorem on sums of two squares. It implies that for an odd prime number , is a prime ideal if and is not a prime ideal if . This, together with the observation that the ideal is prime, provides a complete description of the prime ideals in the Gaussian integers. Generalizing this simple result to more general rings of integers is a basic problem in algebraic number theory. Class field theory accomplishes this goal when K is an abelian extension of Q (that is, a Galois extension with abelian Galois group).
note that because  the ideals generated by and are the same A complete answer to the question of which ideals remain prime in the Gaussian integers is provided by Fermat's theorem on sums of two squares It implies that for an odd prime number  is a prime ideal if and is not a prime ideal if  This together with the observation that the ideal is prime provides a complete description of the prime ideals in the Gaussian integers Generalizing this simple result to more general rings of integers is a basic problem in algebraic number theory Class field theory accomplishes this goal when K is an abelian extension of Q (that is a Galois extension with abelian Galois group)