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Chapter 8: Problem 29

Find a generator of the indicated ideals in the indicated rings. $$ \left\langle x^{2}-5 x+6\right) \cap\left\langle x^{2}-3 x+2\right\rangle \quad \text { in } Q[x] $$

Short Answer

Expert verified
The generator is \( x-2 \).

Step by step solution

01

Understanding the Problem

We are asked to find a generator for the intersection of two ideals in the polynomial ring \( \mathbb{Q}[x] \). The ideals are \( \langle x^2 - 5x + 6 \rangle \) and \( \langle x^2 - 3x + 2 \rangle \).
02

Factor the Polynomials

First, factor the polynomials to find their roots. \( x^2 - 5x + 6 = (x-2)(x-3) \) and \( x^2 - 3x + 2 = (x-1)(x-2) \).
03

Common Divisor of the Ideals

The intersection of the ideals is generated by the polynomial whose roots are the common roots of \( (x-2)(x-3) \) and \( (x-1)(x-2) \). The roots of \( (x-2) \) are in both factorizations.
04

Construct the Generator

The generator is \( x-2 \) because \( x-2 \) is the common factor of both polynomials. Hence, the intersection of the two ideals is generated by \( x-2 \).
05

Verify the Solution

Verify that \( x-2 \) is indeed in both ideals. Since \( x-2 \) divides both \( x^2 - 5x + 6 \) and \( x^2 - 3x + 2 \), it will be a generator for their intersection.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Ring
A Polynomial Ring, typically denoted as \( \mathbb{Q}[x] \) in our example, consists of polynomials with coefficients from the set of rational numbers, \( \mathbb{Q} \). Let's break it down further:
  • Polynomials: These are mathematical expressions involving sums of powers of a variable, usually \( x \), with coefficients that are numbers from the chosen set.
  • Rational Coefficients: In \( \mathbb{Q}[x] \), every coefficient is a rational number, meaning that it can be expressed as a fraction of integers.
  • Operations: Within a polynomial ring, you can add, subtract, and multiply polynomials, adhering to the usual rules of algebra.
Polynomial rings are significant because they allow us to perform algebraic operations and study the properties of polynomials systematically. In the context of our exercise, the polynomial ring \( \mathbb{Q}[x] \) serves as the framework where we define and manipulate the ideals we need to investigate further for their intersection.
Ideal Intersection
An Ideal Intersection involves finding a common generator such that it belongs to multiple ideals within a polynomial ring. In simpler terms, this involves identifying a polynomial that can "fit" into each of the given polynomial expressions seamlessly.
  • Definition of Ideal: An ideal in a ring is a set of elements where the product of any element of the ring with any element in the set is still within the set.

  • Intersection of Ideals: This concept involves finding polynomials that are solutions (or generators) common to all ideals under consideration.

  • Generator of Intersection: The task in our problem was to find such a polynomial which fits perfectly in both ideals given.
In our exercise, the task was simplified by identifying the common factors of the two given polynomials after factoring. The resulting polynomial \( x-2 \) served as the generator, proving it to belong to the intersection of the ideals.
Polynomial Factorization
Polynomial Factorization is the process of expressing a polynomial as a product of simpler polynomials. This technique is crucial in understanding and solving our exercise.
  • Why Factor Polynomials: Finding factors helps in problem-solving by identifying roots and simplifies expressions for further operations such as division and intersection of ideals.

  • Step-by-Step Factorization: Our polynomials were \( x^2 - 5x + 6 \) and \( x^2 - 3x + 2 \).
    • The first polynomial factored to \( (x-2)(x-3) \), indicating its roots are \( x=2 \) and \( x=3 \).
    • The second polynomial factored to \( (x-1)(x-2) \), showing its roots as \( x=1 \) and \( x=2 \).

  • Application in Ideal Intersection: By examining the factors, \( x-2 \) emerged as the common root. Hence, \( x-2 \) became the generator for the intersection, as it divides both initial polynomials.
Factorization not only aids in simplifying polynomials but also plays a pivotal role in identifying common elements for ideal intersections, as illustrated in this exercise.

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Most popular questions from this chapter

In Exercises 1 through 7 (a) Show that the indicated set \(I\) is an ideal in \(Q[x] .(b)\) Find a polynomial \(g(x)\) in \(Q[x]\) such that \(I=\langle g(x)\rangle(c)\) Determine whether \(I\) is a maximal ideal in \(\mathrm{Q}[x]\). \(I=\\{f(x) \in Q[x] \mid f(\sqrt{2})=f(\sqrt{3})=0\\}\)

Determine whether the indicate quotient rings are fields. Justify your answers. $$ \mathrm{Q}[x] /\left\langle x^{2}+3 x+2\right\rangle $$

Determine whether the indicate quotient rings are fields. Justify your answers. $$ \mathbb{Z}_{3}[x] /\left\langle x^{2}+x+1\right\rangle $$

In Exercises 1 through 7 (a) Show that the indicated set \(I\) is an ideal in \(Q[x] .(b)\) Find a polynomial \(g(x)\) in \(Q[x]\) such that \(I=\langle g(x)\rangle(c)\) Determine whether \(I\) is a maximal ideal in \(\mathrm{Q}[x]\). \(I=\\{f(x) \in Q[x] \mid f(2)=f(3)=0\\}\)

Find a generator of the indicated ideals in the indicated rings. $$ \langle x+1\rangle \cap\left\langle x^{2}+1\right\rangle \quad \text { in } \mathbb{Z}_{2}[x] $$
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