In our algebra course our professor said, during a the beginning of a chapter on field extension, that $\left[\mathbb{F}_{p^{k}}:\mathbb{F}_{p}\right]=k$ (where $p$ is obviously prime).
My question is: Why can we even assume that $\mathbb{F}_{p}$ is a subfield of $\mathbb{F}_{p^k}$ ?
$\mathbb{F}_{p}$ isn't closed under the restriction of the multiplication in $\mathbb{F}_{p^{k}}$ to $\mathbb{F}_{p}$, so $\mathbb{F}_{p}$ doesn't form a subfield (for example, for $1,2\in\mathbb{F}_{3}\subseteq\mathbb{F}_{3^{2}}$ we have that $2+1=3\in\mathbb{F}_{3^{2}}$ and not $0$, as we should obtain in $\mathbb{F}_{3}$), so in my opinion we can't even talk about $\left[\mathbb{F}_{p^{k}}:\mathbb{F}_{p}\right]$, since we defined this only for field extension (I realize that we still can talk about $\left[F:G\right]$, if we define this in a more general way just for fields $F,G$ such that $F$ is a $G$-vector space; but in that case I would be annoyed by the slopiness of our course).