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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.

이승우
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    Hint : Assume that the characteristic is composite and derive a contradiction. – Peter Oct 09 '22 at 12:20
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    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 $\mathbb{Q}$. – Vercingetorix Oct 09 '22 at 12:21
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    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. – Vercingetorix Oct 09 '22 at 12:21
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    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 Oct 09 '22 at 12:45

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