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Are there fields other $Z_2$ where there are matrices other than the zero matrix which are both symmetric and anti-symmetric at the same time?

( $Z_2$ is {0,1} with modulo 2 addition and multiplication )

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You are asking for fields with characteristic 2. –  akkkk Dec 3 '12 at 11:15
    
@akkkk: Why did you delete your answer? –  wj32 Dec 3 '12 at 11:17
    
@wj32: I don't know, I wanted to add that the only field satisfying that was $F_2$, but then I realized that's not true so I figured I'm not an expert after all ;) –  akkkk Dec 3 '12 at 11:18
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1 Answer

up vote 2 down vote accepted

Any field with characteristic 2 has this property.

As you may recall, the characteristic of a ring $R$ is the smallest positive number $n$ such that $\sum_1^n 1_R=0$.

Concretely, let $A$ be a matrix over a field of characteristic 2, then $A^T$ also is, but $A^T+A^T=2A^T=0$ so $A^T$ is an additive inverse for $A^T$, which we also denote as $A^T=-A^T$.

Examples of other fields of characteristic 2 are: rational functions over $F_2$, the algebraic closure of $F_2$, and Laurent series over $F_2$.

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What does that mean? –  Robert S. Barnes Dec 3 '12 at 11:19
    
@RobertS.Barnes: It means that $1 + 1 = 0$, where $1$ is the multiplicative identity in the field. –  wj32 Dec 3 '12 at 11:19
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Or to put it another way, a field has characteristic 2 if and only if it has ${\bf Z}_2$ as a subfield. –  Gerry Myerson Dec 3 '12 at 11:44
    
The notation $F_2$ indicates a field of characteristic 2? –  Robert S. Barnes Dec 3 '12 at 12:42
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I'm sorry, no, $F_2$ is the field of order 2, so just what you call $\mathbf{Z}_2$. –  akkkk Dec 3 '12 at 13:33
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