We are only considering subfields of $ℂ$.
Let $K$ be a field and $x$ be transcendental over $K$. The author proves that $[K(x):K]= \infty$ by simply stating "notice that the elements $1, x, x^2, ...$ are linearly independent over $K$."
I understand why such elements are linearly independent over $K$, however I don't see why that implies that $[K(x):K]=\infty$.
I would appreciate any help/thoughts!