Desperate help pls Question 3Let deZ be a square-free integer (that is d#1, and d has no integer factors of the form e except e = t1, cf. p.346). Let R= ZVd = {a + byd| a, b€ Z}. Our ultimate target in this problem is to prove that every prime idealPCR is a maximal ideal. I provide steps below.a) We firstly prove that every ideal IC R is finitely generated. We actually prove a stronger statement that everyideal is generated by at most two elements. The steps provided below are very similar to those of Exercise 32, p.344, which may be helpful.a.1) Prove that if I is non-zero, then Inz is a non-zero ideal in Z.a.2) Derive that there exists a positive integer z E Z such thatInz= (za |a € 2)a.3) Let J be the set of all integers o such that a + bva e I for some a eZ (that is there exists a € Z such thata+ bva e 1). Prove that there exists a positive integer y such thatJ = {yt | t € Z}a.4) Explain why there exista ae Z much that a+ yvāe 1.a.5) Prove that I = {Az + B(s + yvd) | A, B € Z). (That implies that I= (z, s+ yva) in p.145 notations forfinitely generated ideals, but we will not need this fact.) b) Derive from the statements a.1 - a.5 that the factor ring R/P is a finite ring without zero divisors.b.1) Explain why the factor ring R/P has no zero divisors.b.2) Prove that the factor ring R/P is finite.e) Derive from b.1) and 6.2 that R/P is a field.d) Derive from a) - e) that every prime ideal PCR is a maximal ideal.

We are entitled to solve only one question at a time and up to 3 sub parts only . Given - d ∈ℤis a…

Question 3 Let d€Z be a squaze-free integer (that is d# 1, and d has no integer factors of the form ² except e= a1, cf. p. 346). Let R= Z[Va = {a + bva| a, b € Z). Our ultimate target in this problem ia to prove that every prime ideal PCRia maximal ideal. I provide steps below. a) We firstly prove that every ideal ICR is finitely generated. We actually prove a stronger statement that every ideal is generated by at most two elements. The steps provided below are very similar to those of Exercise 32, p. 344, which may be helpful. a.1) Prove that if I is non-aero, then InZ is a non-zero ideal in z. a.2) Derive that there exists a positive integer z € Z such that Inz= (za |a € 2} a.3) Let J be the set of all integers b such that a+ bvd eI for some EZ (that is there exists a €Z such that a+ bva e). Prove that there exists a positive integer y such that J= {xt | 1 € Z}

Question 3 Let d€Z be a squaze-free integer (that is d# 1, and d has no integer factors of the form ² except e= a1, cf. p. 346). Let R= Z[Va = {a + bva| a, b € Z). Our ultimate target in this problem ia to prove that every prime ideal PCRia maximal ideal. I provide steps below. a) We firstly prove that every ideal ICR is finitely generated. We actually prove a stronger statement that every ideal is generated by at most two elements. The steps provided below are very similar to those of Exercise 32, p. 344, which may be helpful. a.1) Prove that if I is non-aero, then InZ is a non-zero ideal in z. a.2) Derive that there exists a positive integer z € Z such that Inz= (za |a € 2} a.3) Let J be the set of all integers b such that a+ bvd eI for some EZ (that is there exists a €Z such that a+ bva e). Prove that there exists a positive integer y such that J= {xt | 1 € Z}

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.4: Maximal Ideals (optional)

Problem 6E

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