Algebraic extension

Question

how to prouve that: every algebraic extension to $k$ is equipotent to a set in $k\times \mathbb{N}$? And if $k$ is infinite feild we have $k$ is equipotent to $K$. My idea is to use this set $G=${$(P,x)∈k[X]^* \times K| P(x)=0$}$.

Thanks a lot,

Answer 1

You're on the right track.

The number of polynomials over $k$ is $|k\times\Bbb N|$, and every polynomial can have only finitely many roots in the extension, so it is still $|k\times\Bbb N|$.