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?

Thank you in advance.

  • 2
    $\begingroup$ Hint : Assume that the characteristic is composite and derive a contradiction. $\endgroup$
    – Peter
    Commented Oct 9, 2022 at 12:20
  • 2
    $\begingroup$ 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 $\mathbb{Q}$. $\endgroup$ Commented Oct 9, 2022 at 12:21
  • 2
    $\begingroup$ 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. $\endgroup$ Commented Oct 9, 2022 at 12:21
  • 2
    $\begingroup$ 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. $\endgroup$
    – 이승우
    Commented Oct 9, 2022 at 12:45


Browse other questions tagged .