Irreducible Polynomial Over the Ring of Polynomials Over Integral Domain

Problems and solutions of ring theory in abstract algebra

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 $S[x]$.

 
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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.)


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