I just need some help understand what an extension field is.
We know $[Q{(\sqrt2}):Q]=2$
Is this the same as saying what is the degree of the min. monic polynomial $p(x)\in Q[x]$ s.t $p(\sqrt2)=0$. Does this same reasoning work for the problem below?
$f(x)\in GF(p)$ where $\deg f(x)=k$ and $x(x^ {p^{n-1}}-1)\in GF(p)$
Then the finite extension $[GF(p)(\alpha):GF(p)]=k$ as $f(\alpha)=0$ where $f(x)\in GF(p)$ with $\deg f(x)=k$
Thank you for the help
I don't know what all the $G(p)$ and $GF(p)$ are, but it is true that for any field $K$ and $\alpha$ algebraic over $K$, the degree of the extension $[K(\alpha):K]$ equals the degree $k$ of the minimal polynomial $f$ of $\alpha$.
This is because of the definition of a minimal polynomial, which implies that $1, \alpha, \alpha^2,\ldots,\alpha^{k-2}, \alpha^{k-1}$ are all linearly independent over $K$ (otherwise $f$ wouldn't be minimal), but $\alpha^k$ lies in their span (because $f(\alpha)= 0$, which may be rearranged into $\alpha^k = \cdots$).