I am looking for an intuitive or example based explanation as to why Galois Fields exist only if the number of elements are a power of a prime. I find most explanations I've read difficult to understand. I am studying this topic from the perspective of Information Theory, and do not wish to delve too deep into the specifics. Thank you.
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A finite field is a vector space over $\Bbb F_p$, so has a basis $(x_1,\ldots ,x_n)$. But then counting gives, since every $x$ is a linear combination of the basis, that we have necessarily $p^n$ elements. So you only need to "delve deep into vector spaces", but this cannot be avoided anyway. – Dietrich Burde May 14 '20 at 13:15
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Do you know what the characteristic of a field is? – mindfields May 14 '20 at 13:19