Every non-singular quadratic form over F2 can be written as an orthogonal sum Am + Bn 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 2n, then it has 22n−1 + 2n−1 elements of norm 1 if its Arf invariant is 1, and 22n−1 − 2n−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.
| 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}}