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In Linear Algebra 1, our professor provided the following theorem:

Let $\Bbb F$ be a finite field. Therefore, there exists some prime number p, and n $\ge$ 1 S.T |$\mathbb F$|= $p^n$.

The proof is left as an exercise to the reader. I know that z/p is a field $\iff p$ is prime, but how does that fact help me prove the theorem? I am stumped.

Noam
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    What have you tried? Alternatively, what is the context behind this problem; where did you find it, and what relevant knowledge do you have? – Christian E. Ramirez Feb 18 '23 at 18:28
  • Hint: let us write $;\Bbb F_p:=\Bbb Z/p;$ for the prime (minimal, if you want) field of characteristic $;p;$ . Then any other field $;\Bbb F;$ containing this field (i.e., any field of characteristic $;p$) is a linear space over $;\Bbb F_p;$ . If $;\dim_{\Bbb F_p}\Bbb F=n;$ , then it is a basic exercise in basic linear algebra to show that $;|\Bbb F|=p^n;$ . Work on this – DonAntonio Feb 18 '23 at 18:29
  • @DonAntonio Showing that |$\mathbb F$|= $p^n$ seems now like more of a counting problem than a linear algebra one? – Noam Feb 18 '23 at 18:46
  • @Noam Well, yes...but you must know the basic linear algebra about unique expression of any vector as linear combination of a basis . It still requires algebra. – DonAntonio Feb 18 '23 at 18:56

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