Let $\mathbb{C}[t]$ be the ring of polynomials over $\mathbb{C}$, and let $\mathbb{C}(t)$ be the quotient field. Let $f(x) = x^3 - x + t \in \mathbb{C}(t)[x]$.
I want to show that $f$ is irreducible. My idea is this: $t$ is a prime in $\mathbb{C}[t]$, so $f$ satisfies Eisenstein's Criterion at the prime $t$, so $f$ is irreducible. Am I applying this correctly? I get confused when we have multiple transcendental variables (like $x,t$ in this case).
The prime $t$ doesn't divide the coefficient of $x$, so the conditions for Eisenstein's criterion are not met.
But $f$ is cubic, hence if it was reducible in $\mathbb{C}(t)[x]$, it would have to have a root, $r$ say, in $\mathbb{C}(t)$.
Since the coefficients of $f$ are in $\mathbb{C}[t]$ which is a UFD, the rational root test can be applied.
Since $f$ is monic and has constant term $t$, it follows that $r \in \mathbb{C}[t]$ and $r$ divides $t$ in $\mathbb{C}[t]$.
Since $t$ is prime in $\mathbb{C}[t]$, hence either $r = a\,$ or $r=at$, for some nonzero $a \in \mathbb{C}$.
But
$$f(a) = a^3 - a + t$$
$$\text{and}$$
$$f(at) = a^3t^3 - at + t$$
both of which are nonzero elements of $\mathbb{C}(t)$.
It follows that $f$ is irreducible in $\mathbb{C}(t)[x]$.