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invariant of Arf

An invariant of a quadratic form modulo 2, given on an integral lattice endowed with a bilinear skew-symmetric form. Let be an integral lattice of dimension and let be a form for which . There exists bases , called symplectic bases, in which the matrix of reduces to block-diagonal form: The diagonal contains the blocks


while the other entries are zero.

Suppose that a mapping

is given on such that

(a "quadratic form modulo 2" ). The expression

is then called an Arf-invariant [1]. If this expression equals zero, then there is a symplectic basis on all elements of which vanishes; if this expression equals one, then there is a symplectic basis on all elements of which, except and , the form vanishes, while


[1] C. Arf, "Untersuchungen über quadratischen Formen in Körpern der Charakteristik 2, I" J. Reine Angew. Math. , 183 (1941) pp. 148–167


See [a1], Appendix 1, for material concerning the Arf-invariant for inner product spaces over a field of characteristic 2 in relation to the Witt algebra of quadratic inner product spaces over .


[a1] J. Milnor, D. Husemoller, "Symmetric bilinear forms" , Springer (1973)
How to Cite This Entry:
Arf-invariant. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article