Page 516 of Dummit and Foote's Abstract Algebra says:
Let $F=k(t)$ be the field of rational functions in the variable $t$ over a field $k$ (for example, $k=\mathbb{Q}$ or $k=\mathbb{F}_{p}$). Let $p(x)=x^{2}-t\in F[x]$. Then $p(x)$ is irreducible (it is Eisenstein at the prime $(t)$ in $k[t]$).
My questions: how did they apply Eisenstein's Criterion (I'm assuming that's what they're talking about when they say Eisenstein) to $p(x)$? Why is $(t)$ prime? (I didn't even know you could use Eisenstein's Criterion for primes besides the integer primes, but I don't know how to prove it, so maybe the explanation is there.) Would we want to show that $(t)$ divides $t$ but $(t)^{2}$ doesn't divide $t$? . . .
Also, just to make sure I understand, they said $F$'s elements are "rational functions in the varible $t$," so does that mean $p(x)$ is a polynomial in $x$ coefficients are rational functions in are rational functions in the varible $t$?
Thanks in advance.