Purchase Solution

Maximal Ideals of Prime and Primitive Polynomials

Not what you're looking for?

Suppose that p is a prime and f is a primitive polynomial in Z[x] which is irreducible modulo p. Show that the set of maximal ideals in Z[x] all can be written as (p,f). Also show that there does not exist a principal ideal in Z[x] which is maximal.

Solution Summary

This solution helps show that a set of maximal ideals in a function can be written in a certain way.

Solution Preview

Let M be a maximal ideal of Z[x].
Let us first show that M∩Z≠{0}. To this end, assume that M∩Z={0} . Let N be the ideal of Q[x ] generated by M. Since Z[x]≠Q[x], N≠Q[x]. Since Q is a field, every proper ideal of Q[x] is principal. Therefore, N=(f(x)) , where f(x)∈Q[x ] and degf(x)>0. Without loss of generality, we may assume that f(x)∈Z[x], and content of f(x) is 1. Let us show that M=(f(x)) in Z[x]. Indeed, consider g(x) from M. Then g(x)∈N, and hence g(x)=f(x)h(x), for some h(x)∈Q[x]. Gauss lemma implies that h(x) ∈Z[x]. This implies that M=(f(x)) in Z[x].
Let us now show that Z[x]/(f(x)) is not a field. Since deg ...

Probability Quiz

Some questions on probability

Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.