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Chapter 17: Problem 16

If \(R\) is an integral domain, prove that if \(f(x)\) is a unit in \(R[x]\), then \(f(x)\) is a constant and is a unit in \(R\).

Short Answer

Expert verified
If \(f(x)\) is a unit in \(R[x]\), then \(f(x)\) is a constant polynomial and \(f(x)\) is a unit in the integral domain \(R\).

Step by step solution

01

Define What It Means for a Polynomial to Be a Unit

A polynomial \(f(x)\) is a unit in the polynomial ring \(R[x]\) if there exists another polynomial \(g(x)\) such that the product of \(f(x)\) and \(g(x)\) is equal to 1, i.e., \(f(x)g(x) = 1\). This definition is analogous to the definition of a unit element in a ring.
02

Argue That the Polynomial Is a Constant

Assume that \(f(x)\) is a unit in \(R[x]\). Then there exists a polynomial \(g(x)\) such that \(f(x)g(x) = 1\). The polynomial 1 is a constant polynomial, and the degree of the product of two polynomials is the sum of their degrees. As the degree of 1 is 0, then \(f(x)\) and \(g(x)\) must be zero degree polynomials, i.e., constant polynomials. Hence \(f(x)\) is a constant polynomial.
03

Show That the Polynomial Is a Unit in R

Since \(f(x)\) is a constant polynomial, we can write it as \(f(x) = c\) for some \(c\) in \(R\). If \(f(x)\) is a unit in \(R[x]\), then the constant \(c\) must be a unit in \(R\) because it has a multiplicative inverse in \(R\), which is the constant term of the polynomial \(g(x)\). Therefore, if \(f(x)\) is a unit in \(R[x]\), then \(f(x)\) is a constant and is a unit in \(R\).

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Most popular questions from this chapter

Does the Division Algorithm (for polynomials) hold in the integral domain Z \(\\{x]\) ? Explain.

Let \(f(x), g(x) \in \mathbf{R}[x]\) with \(f(x)=x^{3}+2 x^{2}+a x-b, g(x)=x^{3}+x^{2}-b x+a\). Determine values for \(a, b\) so that the gcd of \(f(x), g(x)\) is a polynomial of degree 2 .

a) If \(f(x)=x^{4}-16\), find its roots and factorization in \(Q[x]\). b) Answer part (a) for \(f(x) \in \mathbf{R}[x]\). d) Answer parts (a), (b), and (c) for \(f(x)=x^{4}-25\). c) Answer part (a) for \(f(x) \in \mathbf{C}[x]\).

Let \(p\) be a prime. a) How many monic quadratic (degree 2) polynomials \(x^{2}+b x+c\) in \(\mathbf{Z}_{p}[x]\) can we factor into linear factors in \(\mathbf{Z}_{\rho}[x]\) ? (For example, if \(p=5\), then the polynomial \(x^{2}+2 x+2\) in \(Z,[x]\) would be one of the quadratic polynomials for which we should account, under these conditions.) b) How many quadratic polynomials \(a x^{2}+b x+c\) in \(\mathbf{Z}_{p}[x]\) can we factor into linear factors in \(\mathbf{Z}_{p}[x]\) ? c) How many monic quadratic polynomials \(x^{2}+b x+c\) in \(\mathbf{Z}_{p}[x]\) are irreducible over \(\mathbf{Z}_{p} ?\) d) How many quadratic polynomials \(a x^{2}+b x+c\) in \(\mathbf{Z}_{p}[x]\) are irreducible over \(\mathbf{Z}_{p}\) ?

\text { Given a }(v, b, r, k, \lambda) \text {-design with } b=v \text {, prove that if } v \text { is even, then } \lambda \text { is even. }
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