Is a finite field algebraic extension of a fixed degree unique or just unique up to isomorphic?

Question

well,if $[F_{q}(\alpha):F_{q}]=m$, can i prove that $F_{q}(\alpha)=F_{q^{m}}$ ? or i can only prove that $F_{q}(\alpha)$ is isomorphic to $F_{q^{m}}$ ?

thanks for advance.

Answer 1

$=$ can be sort of slippery. You really want $\cong$. For example, $\mathbb F_5[x] / (x^2+2)$, and $\mathbb F_5[x] / (x^2+3)$ are both $F_{25}$, but they're not "equal", they are different (but isomorphic) quotients of the same ring.