| Arf Invariant |
Article Index for Arf |
Information AboutArf Invariant |
| CATEGORIES ABOUT ARF INVARIANT | |
| quadratic forms | |
|
In Mathematics , the Arf invariant, named after Cahit Arf , who introduced it in 1941, is an element of '''F'''2 associated to a non-singular Quadratic Form over the field '''F'''2 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. STRUCTURE 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. REFERENCES |
|
|