I think people would normally call that a transcendental extension and not an algebraic one.
ℚ[π] is not an algebraic extension of ℚ.
I guarantee that is not an algebraic extension.
It's not even a finite extension
That's not a valid critique, as other commenters explained
Finite implies algebraic. The other comment says that the converse isn't true.
You're totally right, I could have worded that differently. I actually posted an infinite field extension that is algebraic elsewhere in this thread, but usually those are tricky and I haven't seen them pop up as often as finite algebraic extensions