Possible Cardinality of a Field

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Solution 1

In priciple, yes. $F$ is a vector space over $\mathbb F_p$ and hence in the finite case it is in bijection with some $\mathbb F_p^n.$ Of course $|\mathbb F_p^n|=|\mathbb F_p|^n=p^n$.

Solution 2

I did not manage to follow your argument however there is a very simple argument here:

Since you already noted that the prime field of $F$ is $\mathbb{F}_{p}$ (up to isomorphism) all you have to recall is that $F$ is a vector space over its prime field hence $$|F|=|\mathbb{F}_{p}|^{dim_{\mathbb{F_p}}(F)}=p^{dim_{\mathbb{F_p}}(F)}$$

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Updated on September 29, 2020

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  • AsinglePANCAKE
    AsinglePANCAKE about 3 years

    The following question struck me as pretty interesting: Let $\Bbb F$ be a field of characteristic $p$ (a prime, of course). I'm then asked to show that $|\mathbb{F}| = p^n$ for some $n\geq 1$.

    Here's my intuition. Certainly we know that the prime subfield of $\mathbb{F}$ has order $p$. Now if there's an element (treating $\mathbb{F}$ now as a vector space over itself) independent from it, we have the $Span\{1,a_1\}$ as the usual set of linear combinations of $1$ and $a_1$. And any element of a field of characteristic $p$ added to itself $p$ times is $0$, so now we have $p^2$ possible linear combinations. And so on, arguing inductively. Is this argument kosher? Or does more need to be said to make it rigorous?

    • Gerry Myerson
      Gerry Myerson about 11 years
      The question is answered math.stackexchange.com/questions/53877/… and elsewhere on this site.
    • AsinglePANCAKE
      AsinglePANCAKE about 11 years
      @GerryMyerson Might you point me to the elsewhere? That answer uses language a bit over my head unfortunately...Apologies for the reposting...
    • Gerry Myerson
      Gerry Myerson about 11 years
      math.stackexchange.com/questions/183462/…? which you can find by looking at the list headed Linked on the right side of the page in my earlier comment. Look around, check out some of the links, you won't break anything.
    • AsinglePANCAKE
      AsinglePANCAKE about 11 years
      @GerryMyerson Except perhaps the patience of veteran stack-exchangers! Many thanks!
    • lhf
      lhf about 11 years
      You mean, a finite field.
  • AsinglePANCAKE
    AsinglePANCAKE about 11 years
    If you don't mind, might you elaborate a bit on the string of equalities? They don't resonate with my n00b-self as obvious...
  • Belgi
    Belgi about 11 years
    Do you know that up to isomorphism there is only one vector space of dimesnion $n$ over a given field ?
  • AsinglePANCAKE
    AsinglePANCAKE about 11 years
    Umm, this makes sense, but I don't think I've seen it before.
  • AsinglePANCAKE
    AsinglePANCAKE about 11 years
    Ah yes, that's the answer I was poorly articulating above. Thank ya!