Arf Invariant - Quadratic Forms Over F₂

Quadratic Forms Over F₂

Over F₂ the Arf invariant is 0 if the quadratic form is equivalent to a direct sum of copies of the binary form, and it is 1 if the form is a direct sum of with a number of copies of .

William Browder has called the Arf invariant the democratic invariant because it is the value which is assumed most often by the quadratic form. Another characterization: q has Arf invariant 0 if and only if the underlying 2k-dimensional vector space over the field F₂ has a k-dimensional subspace on which q is identically 0 – that is, a totally isotropic subspace of half the dimension; its isotropy index is k (this is the maximum dimension of a totally isotropic subspace of a nonsingular form).