Is every field a field extension of some form.

Question

I am new to finite field theory .While I was going through the theory I figured that $\mathbb{C}$ is in fact $\mathbb{R}(i) $ isomorphic to $\mathbb{R}[x]/(x^2+1)$ .So I had a question in mind is every field a form of extension of some other field?

Answer 1

Fields that are not proper extensions are called prime fields and these are exactly $\Bbb Q$ and the finite fields $\Bbb F_p$ for prime $p$.

Every other field $K$ is an extension of exactly one of the prime fields, and depending on which of these, $K$ is said to be of characteristic $p$ (in the case $\Bbb F_p$) or $0$ (in the case $\Bbb Q$).