nonzero element of the ideal of minimal norm. Since the norm is multiplicative, the norm of a generator of one of the ideal factors of (p) must be a strict divisor of , so that we must have , which gives Fermat's theorem. 2. Second proof. This proof builds on Lagrange's result that if is a prime number, then there must be an integer m such that is divisible by p (we can also see this by Euler's criterion); it also uses the fact that the Gaussian integers are a unique factorization domain (because they are a Euclidean domain). Since does not divide either of the Gaussian integers and (as it does not divide their imaginary parts), but it does divide their product , it follows that cannot be a prime element in the Gaussian integers. We must therefore have a nontrivial factorization of p in the Gaussian integers, which in view of the norm can have only two factors (since the norm is multiplicative, and , there can only be up to two factors of p), so it must be of the form for some integers and . This immediately yields that . Proof by Minkowski's Theorem For congruent to mod a prime, is a quadratic residue mod by Euler's criterion. Therefore, there exists an integer such that divides . Let be the standard basis elements for the vector space and set and . Consider the lattice . If then . Thus divides for any . The area of the fundamental parallelogram of the lattice is . The area of the open disk, , of radius centered around the origin is . Furthermore, is convex and symmetrical about the origin. Therefore, by Minkowski's theorem there exists a nonzero vector such that . Both and so . Hence is the sum of the squares of the components of . Zagier's "one-sentence proof" Let be prime, let denote the natural numbers (with or without zero), and consider the finite set of triples of numbers. Then has two involutions: an obvious one whose fixed points correspond to representations of as a sum of two squares, and a more complicated one, which has exactly one fixed point . This proves that the cardinality of is odd. Hence, has also a fixed point with respect to the obvious involution. This proof, due to Zagier, is a simplification of an earlier proof by Heath-Brown, which