Is there an infinite field F with char(F)=p and not algebraically closed field?

Question

Is there an infinite field F with characteristic of the field $F$ is $p$ (p is prime) and not algebraically closed field ?

Answer 1

$\mathbb{F}_p (t)$ is an example. It is defined as

$$ \mathbb{F}_p(t) = \{\frac{f(t)}{g(t)}: f(t),g(t)\ne 0 \in\mathbb{F}_p[t] \} $$

This is the field of rational functions with coefficients in $\mathbb{F}_p$. It is obvious that $\mathbb{F}_p (t)$ is infinite. However, it has characteristic $p$. This is because it contains subfield $\mathbb{F}_p$ and $char(\mathbb{F}_p) = p$. Also, if we consider $X^2 - t$, we see that it is not algebraically closed.

Answer 2

$\Bbb{F}_p(t)$ is a perfect example of such a field.

If you look for an algebraic extension over $\Bbb{F}_p$, I suggest you to consider the following: if $\Omega_p$ denotes the algebraic closure of $\Bbb{F}_p$, any infinite proper subfield of $\Omega_p$ satisfies this condition. For example, we can consider $$K= \{ \alpha \in \Omega_p | [\Bbb{F}_p(\alpha ): \Bbb{F}_p] \mbox{ is a power of }2 \}$$