For a transcendental extension $K(\alpha):K$ for sub-field $K$ of $\mathbb{C}$, $[K(\alpha):K]=\infty$

Showing that the basis for $K(\alpha)$ that describes it as a vector space is infinite leads us nowhere since it would deal with infinite dimensional vector spaces.

So suppose that the degree of the trascendental extension is finite, say $N$. Then if $B$ is the basis for $K(\alpha)$ as a vector space, $|B|=N$ by definition. I do not know how to proceed from here and how to relate to the transcendence of $\alpha$ over $K$.

The book I am working on says that it is enough to show that $1,\alpha,\alpha^2,...$ is linearly independent. But I do not know how to use this without dealing with polynomials of infinite degree (which are not necessarily polynomials).

Any help would be much appreciated!