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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.

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Number of Skew Symmetric Matrices of fixed rank

The number of symmetric matrices of order $n$ and rank $r$ over finite fields has been counted e.g. http://www.math.clemson.edu/~kevja/REU/2004/SymmetricRankRMatrices.pdf Is the number of skew-symm …
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