Why can't the polynomials have a root in $K$?

Question

I want to show the following statement:

Each proper subfield of an algebraically closed field is not algebraically closed.

$$$$

Suppose that it doesn't hold. Let $F$ be an algebraically closed field and $K$ a proper subfield of $F$.

Suppose that $K$ is algebraically closed. That means that all the non-constant polynomials of $K[x]$ have a root in $K$.

Why is this impossible?

Answer 1

It's false, the set of complex numbers that are a root to a polynomial with integer coefficients form an algebraic closed field that does not contain $\pi$.