4. Let p be a rational prime.¹(a) Prove that p factors nontrivially (i.e., does not stay prime) in Z[i] if and only if thereare integers a and b such that p = a² + b². (Hint: what must the norm of a nonunitfactor of p in Z[i] be?)(b) Deduce from (a) that, if p factors nontrivially in Z[i], then p = 2 or p = 1 (mod 4).(Hint: for odd p, use part (b) of the previous problem.)

a)Assume that p factors nontrivially in Z[i]. This means p can be expressed as the product of two…

Let p be a rational prime.¹ (a) Prove that p factors nontrivially (i.e., does not stay prime) in Z[i] if and only if there are integers a and b such that p = a² +6². (Hint: what must the norm of a nonunit factor of p in Z[i] be?) (b) Deduce from (a) that, if p factors nontrivially in Z[i], then p = 2 or p = 1 (mod 4). (Hint: for odd p, use part (b) of the previous problem.)

Let p be a rational prime.¹ (a) Prove that p factors nontrivially (i.e., does not stay prime) in Z[i] if and only if there are integers a and b such that p = a² +6². (Hint: what must the norm of a nonunit factor of p in Z[i] be?) (b) Deduce from (a) that, if p factors nontrivially in Z[i], then p = 2 or p = 1 (mod 4). (Hint: for odd p, use part (b) of the previous problem.)

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.2: Properties Of Division

Problem 52E

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