to a sum of sauar o have now AY auesion for the only even prime (Fermat's p = a² + b² Theorem) Let p be an odd prime in Z. Then p = a² + b² for integers a and b in Z if and only if p = 1 (mod 4)....

Section 47 number 10 (a)and (b). Use theorem 47.10

Extracted text: to a sum of sauar o have now AY auesion for the only even prime (Fermat's p = a² + b² Theorem) Let p be an odd prime in Z. Then p = a² + b² for integers a and b in Z if and only if p = 1 (mod 4). 47.10 Theorem


Extracted text: of D. among all |N(B)| > 1 for BE D.Show that n is an irreducible 10 a. Show that 2 is equal to the product of a unit and the square of an irreducible in Z[i]. b. Show that an odd prime p in Z is irreducible in Zliif and only if p = 3 (mod 4). (Use Theorem 47.10.) 1 Prove Lemma 47.2. Duun thot N of Fxamnle 47 9 is multiplicative, that is, that N(aß) = N(a)N(B) for a, BE Z[v-5).