# Irreducible Polynomial Over the Ring of Polynomials Over Integral Domain

## Problem 333

Let $R$ be an integral domain and let $S=R[t]$ be the polynomial ring in $t$ over $R$. Let $n$ be a positive integer.

Prove that the polynomial
$f(x)=x^n-t$ in the ring $S[x]$ is irreducible in $S[x]$.

## Proof.

Consider the principal ideal $(t)$ generated by $t$ in $S$.
Then the ideal $(t)$ is a prime ideal in $S$ since the quotient
$S/(t)=R[t]/(t)\cong R$ is an integral domain.

The only non-leading coefficient of $f(x)=x^n-t$ is $-t$, and $-t$ is in the ideal $(t)$ but not in the ideal $(t)^2$.
Then by Eisenstein’s criterion, the polynomial $f(x)$ is irreducible in $S[x]$.

(Remark that $S=R[t]$ is an integral domain since $R$ is an integral domain.)

Let $R$ be a ring with unity. Suppose that $f$ and $g$ are ring homomorphisms from $\Q$ to $R$ such...