Definitions
The Arf invariant belongs to a quadratic form over a field K of characteristic 2. Any binary non-singular quadratic form over K is equivalent to a form with in K. The Arf invariant is defined to be the product . If the form is equivalent to, then the products and differ by an element of the form with in K. These elements form an additive subgroup U of K. Hence the coset of modulo U is an invariant of, which means that it is not changed when is replaced by an equivalent form.
Every nonsingular quadratic form over K is equivalent to a direct sum of nonsingular binary forms. This has been shown by Arf but it had been earlier observed by Dickson in the case of finite fields of characteristic 2. The Arf invariant Arf is defined to be the sum of the Arf invariants of the . By definition, this is a coset of K modulo U. Arf has shown that indeed Arf does not change if is replaced by an equivalent quadratic form, which is to say that it is an invariant of .
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.
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