Why is $\left[\mathbb{R}(t):\mathbb{R} \right]$ infinite?

Question

When I was reading about extensions over $\mathbb{R}$, I've saw somebody affirming this. Why is this true? How can I prove it? Thank you!

Answer 1

$1, t, t^2, \dots, t^n \dots$ is a linearly independent family of polynomials of $\mathbb R[t]$. And $\mathbb R[t] \subsetneq \mathbb R(t)$.