Question

Theorem 16.1. Let p be a prime number. Suppose r is a Gaussian integer satisfying N(r) = p. Then r is irreducible in Z[i]. In particular, if a and b are integers such that a² +62 = p, then the Gaussian integers Ea – bi and £b£ai are irreducible. Exercise 16.1. Prove Theorem 16.1. (Hint: For the first part, suppose st is a factorization of r. You must show that this factorization is trivial. Apply the norm to obtain p= N(s)N(t). Deduce that either N(s) or N(t) equals 1 and conclude that either s or t is a unit.)

Answer 1

Theorem 16.1. let, p ke kime, & be a Gaussian integer a N(M)=b. Then, p is irreducible in Z E . moreover, if a, b € 72 2 a +b=p, then, zatib, tbtia are irreduc Prof. ler un anume the contrary .. Say, para r be a Gaunian inregen and n is not irreducible. it. I m, na p= mn where . neither mhorn is a unit in. N(m) ² tl ,N(W) + +1. therefore, p=mn > N(M) = N(m) N(n) = b = NW) NIN) and we have W(m) and (h) can not be 21. hence, I two non-unit integers say ab e abap and N(m) =a & {11} N(n) = 6€ 113 > p has two now unit divisors, namely alb > p is not fine.. (E) hence, PER[i] is irreducible. let, pe {atib, Q-ib, ratib, -Q-ib, bria, bria, bta zb-ia? cleanly, N(n) = a + b²=p, a kime. by the above logical consequence, we can achieve that -n is irreducible in Zh. hemu all the elements o the forms tazib or abtia are irreducible in Z[i]. O