- Arf invariant
In
mathematics , the Arf invariant, named after Turkishmathematician Cahit Arf , who introduced it in 1941, is an element of F2 associated to a non-singularquadratic form over the field F2 with 2 elements, equal to the most common value of the quadratic form. Two such quadratic forms are isomorphic if and only if they have the same Arf invariant.tructure of quadratic forms
Every non-singular quadratic form over F2 can be written as an orthogonal sum "A""m" + "B""n" of copies of the two 2-dimensional forms "A" and "B", where "A" has 3 elements of norm 1, and "B" has one element of norm 1. The numbers "m" and "n" are not uniquely determined, because "A" + "A" is isomorphic to "B" + "B". However "m" is uniquely determined mod 2, and the value of "m" mod 2 is the Arf invariant of the quadratic form.
If "B" is a quadratic form of dimension 2"n", then it has 22"n"−1 + 2"n"−1 elements of norm 1 if its Arf invariant is 1, and 22"n"−1 − 2"n"−1 elements of norm 1 if its Arf invariant is 0.
The Arf invariant is additive; in other words, the Arf invariant of an orthogonal sum of two quadratic forms is the sum of their Arf invariants.
ee also
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Arf-Kervaire invariant References
* citation
first = Arf|last= Cahit
authorlink = Cahit Arf
title = Untersuchungen über quadratische Formen in Körpern der Charakteristik 2, I
journal = J. Reine Angew. Math
volume = 183
year = 1941
pages = 148–167*
last = Kirby|first= Robion
authorlink = Robion Kirby
title = The topology of 4-manifolds
year = 1989
series = Lecture Notes in Mathematics|volume= 1374|publisher= Springer-Verlag
ISBN =0-387-51148-2
doi=10.1007/BFb0089031
*springer|id=A/a013230|title=Arf invariant|author=A.V. Chernavskii
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