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Chapter 17: Problem 10

If \(F\) is any field, let \(f(x), g(x) \in F[x]\). If \(f(x), g(x)\) are relatively prime, prove that there is no element \(a \in F\) with \(f(a)=0\) and \(g(a)=0\).

Short Answer

Expert verified
So, it's impossible for an element \(a \in F\) to be a zero of both \(f(x)\) and \(g(x)\) when they're relatively prime.

Step by step solution

01

Understand The Definitions

A pair of polynomials \(f(x)\) and \(g(x)\) are said to be relatively prime if their greatest common divisor (GCD) is 1. A zero of a polynomial is an element \(a\) in a field \(F\) such that the polynomial evaluated at \(a\) is equal to zero. In other words, if \(f(a) = 0\), \(a\) is a zero of the polynomial \(f(x)\).
02

Contradiction Assumption

Assume to the contrary that there exists an element \(a \in F\) such that \(f(a) = 0\) and \(g(a) = 0\). The goal is to show that this leads to a contradiction.
03

Invoke the Property of Relatively Prime Polynomials

If \(f(x)\) and \(g(x)\) are relatively prime, by Bezout's identity, there exist polynomials \(s(x), t(x)\) such that \(s(x)f(x) + t(x)g(x) = 1\). This identity holds for all \(x \in F\).
04

Contradiction Revealed

Substitute \(x = a\) into the identity from Step 3. This gives \(s(a)f(a) + t(a)g(a) = 1\). But, \(f(a) = 0\) and \(g(a) = 0\) by the assumption made in Step 2. Substituting these values into the equation yields \(0 = 1\), which is a contradiction.

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Most popular questions from this chapter

a) Show that \(s(x)=x^{2}+1\) is reducible in \(\mathbf{Z}_{2}[x]\). b) Find the equivalence classes for the ring \(Z_{2}[x] /(s(x))\). c) Is \(\mathbf{Z}_{2}[x] /(s(x))\) an integral domain?

Let \((R,+, \cdot)\) be a ring. If \(I\) is an ideal of \(R\), prove that \(I[x]\), the set of all polynomials in the indeterminate \(x\) with coefficients in \(I\), is an ideal in \(R[x]\).

Give an example of a polynomial \(f(x) \in \mathbf{R}[x]\) where \(f(x)\) has degree 6 , is reducible, but has no real roots.

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\text { Determine all of the polynomials of degree } 2 \text { in } \mathbf{Z}_{2}[x] \text {. }
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