Finite extensions of $\mathbb F_p(t)$

Question

Let $\mathbb F_p(t)$ the field of rational functions with coefficients in $\mathbb F_p$.

Is it true or not that every finite extension $K$ of $\mathbb F_p(t)$ is $K\cong\mathbb F_{p^m}(t)$ for some $m\ge 1$?

Answer 1

The answer is no. For example, you can consider the fraction field of $\Bbb{F}_p[x,t]/(x^2-t)$: this is a quadratic extension of $\Bbb{F}_p(t)$, where the element $x= \sqrt{t}$ is added.

Answer 2

No. Other such extensions are fields of functions on some curve. For that to make sense you need to know a few basics from algebraic geometry. The idea is that the field of rational functions corresponds to the line, but more complicated curves have more complicated function fields.

For example, if $p>3$, then the equation $$ u^2=t^3+at+b,\qquad(*) $$ with $a,b$ some constants from $\Bbb{F}_p$, chosen in such a way that the cubic has no multiple roots, defines an elliptic curve $E$. The related function field is simply the algebraic extension $\Bbb{F}_p(E)=\Bbb{F}_p(t)[u]$, where $(*)$ gives the minimal polynomial of $u$ over $\Bbb{F}_p(t)$.