Field extension stuff

[Zareon]

I`m trying to prove the following:

 

An algebraically closed field is of infinite extension over its prime subfield.

 

I'm not sure I translated it correctly. What I mean is: Let K be an algebraically closed field and k its prime subfield. Then the degree of the extension K/k is infinite, i.e. [K:k] is not finite.

 

I have in my mind an example like C/Q (the complex and rational number fields). I know it's true in this case, because Q has transcendental numbers.

So I tried to prove k must have a transcendental element, since if the extension were finite, it would be algebraic. But I`m not sure this would work, because not all infinite extensions are algebraic. In any case, I haven't progressed much.

 

Help is appreciated.