Showing degree of algebraic closure is infinite

Question

$R$ is a UFD, but not a field, and $K=Quot(R)$. Show that $[\bar{K}:K]=\infty$ where $\bar{K}$ is the algebraic closure of K.

I am not sure how to approach this problem. Any hint is appreciated. Thanks

Answer 1

Since $R$ is a UFD different of a field, you have an element $p$ which is irreducible, thus $p$ is a prime. The Eisenstein criterion applied to integral domain: https://en.wikipedia.org/wiki/Eisenstein's_criterion#Generalized_criterion

implies that $X^n-p, n>1$ is irreducible in $R[X]$. This implies that $X^n-p$ is irreducible in $K[X]$ see the same reference. Thus for each $n>0$, $F$ as an extension of degree $n$ $F[X]/(X^n-p)$ so its algebraic closure is infinite since if $F$ is finite of cardinal $f$, $F[X]/(X^n-p)$ is a $F$-vector space of dimension $F$ which has dimension $f^n$.