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Diagonalize a form over the rationals to

\mathop{\rm diag}[p^a\cdot A, p^b\cdot B, \ldots],

where all the entries are Integers and $A$, $B$, ...are Relatively Prime to $p$. Then the $p$-signature of the form (for $p\not= -1, 2$) is

p^a+p^b+\ldots+4k {\rm\ (mod\ } 8),

where $k$ is the number of Antisquares. For $p=-1$, the $p$-signature is Sylvester's Signature.

See also Signature (Quadratic Form)

© 1996-9 Eric W. Weisstein