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The units of $R [x]$ over an integral domain are exactly the constant polynomials whose values are units of $R$

Statement

Let $R$ be an integral domain. A polynomial $f \in R [x]$ is a unit if and only if it is a constant polynomial whose constant value is a unit of $R$ .

Facts & Assumptions

Given: An integral domain $R$ and a polynomial $f \in R [x]$ .

[L1]

For nonzero polynomials over a domain, $\deg (f g) = \deg f + \deg g$ (Over an integral domain, degrees add under multiplication of nonzero polynomials).

[L2]

The constant-polynomial map is an injective unital ring homomorphism $R \to R [x]$ (Polynomial convolution makes $R [x]$ a commutative ring containing $R$ as its constant subring).

Proof

technique · direct

1.1

If $f$ is a unit, choose $g$ with $f g = 1$ ; neither factor is zero and [L1] gives $0 = \deg 1 = \deg f + \deg g$ , so both degrees are $0$ , and comparison of constant coefficients shows that the constant value of $f$ is a unit of $R$ .

givenL1L2choose

2.1

Conversely, if $u \in R$ is a unit with inverse $v$ , then [L2] gives $c (u) c (v) = c (u v) = c (1) = 1$ , so the constant polynomial $u$ is a unit of $R [x]$ .

L2algebra∎

Depends on

Used by

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

James McKernan, MIT 18.703 Lecture 21, Lemma 21.1 (standard reference, not scraped)

The units of R[x]R[x] over an integral domain are exactly the constant polynomials whose values are units of RR