# Why is every finite field is a finite extention of a prime field isomorphic to $\mathbb{Z}_p$? [duplicate]

I am studying about finite field and I found my textbook says that "Every finite field is a finite extention of a prime field isomorphic to $$\mathbb{Z}_p$$".

I cannot find a direct proof of this statement.

I think it is because 'every field contains $$\mathbb{Z}_p$$ or $$\mathbb{Q}$$', but I want to find more rigorous proof.

Could you give me a proof of the statement?

• Hint : Assume that the characteristic is composite and derive a contradiction. Commented Oct 9, 2022 at 12:20
• The prime subfield is simply the subfield generated by $1$, i.e, $<1>$. Then $<1> = \mathbb{Z}_p$ where $p$ is the characteristic of the field. If the characteristic is infinite/$0$ we get $\mathbb{Q}$. Commented Oct 9, 2022 at 12:21
• The only remaining thing to see is that the characteristic of a field must be a prime, which is easy enough and I'll let you figure it out. Commented Oct 9, 2022 at 12:21
• Thank you for kind answers. I've got enough intuition from the hint and explanations, and Anne Bauval's link is exactly what I was searching for. Thank you all.
– 이승우
Commented Oct 9, 2022 at 12:45