4. This problem is the consummation of all the themes and topics we have covered up to Quadratic Reci- procity. It constitutes one of the major results of Number Theory of the times of Fermat, Euler, Gauss, etc. The following result was conjectured by Fermat: Let p E 1 (mod 4) be a prime number. Then it is the sum of two squares, that is, there are integers :1: and y with p = :r:2 +y2. It is fair to say that this result and its proof constitutes historically the start of algebraic number theory. Fermat didn't provide a proof. The rst lcnown proof is due to Euler. The one we explore below is from Dedekind. In this problem p is a prime number with p E 1 (mod 4). (a) (1 point) Use problem 1 of this homework to conclude that there exists integers X and K with X2+1=pK {b} (2 points) Factoring the left hand side in the Gaussian integers we have (X+)(X 'i) = pK. Prove that p does not divide, in the Gaussian Integers, either of X +33 or X i. [c] (2 points) Justify carefully why part {b} implies p is not a prime in the Gaussian Integers. (d) (1 point) Justify carefully why.r part (c) implies there exist two Gaussian integers a. + bi and 6+ dz", that are not units, with p = (a + trie + (it). [e] (1 point) Prove that zmkmm+y (f) (2 points) Prove that p = a2 + b2. (g) (1 point) Explain how this result, together with other facts we have seen in the course, complete the proof of the following fact: An odd prime number is a sum of two squares if and only if p E 1 (mod 4)