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A finite field is a field with a finite number of elements. For each prime power $q^k$, there is a unique (up to isomorphism) finite field with $q^k$ elements. Up to isomorphism, these are the only finite fields.

3 votes

Square-free sets in $\mathbb F_2^n\oplus\mathbb F_2^n$

Let $d_n$ be the maximal density of a squarefree set in $\mathbb F_2^n\oplus\mathbb F_2^n$. Then it is unknown whether there exists a constant $c < 1$ such that $d_n = O(c^n)$. Indeed, such a result w …
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14 votes
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Can you use Chevalley‒Warning to prove existence of a solution?

Let $f_i \in \mathbb{F}_q[X_1,\dots,X_n]$ have degree at most $d$ for each $i \in [|1,r|]$, and assume that the affine scheme $$ X = \operatorname{Spec} \mathbb F_q [x_1,\ldots,x_n]/(f_1, \ldots, f_r) …
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