Pfister form

In mathematics, a Pfister form is a particular kind of quadratic form over a field "F" (whose characteristic is usually assumed to be not 2), introduced by A. Pfister in 1965. A Pfister form is in 2^"n" variables, for some natural number "n" (also called an n-Pfister form), and may be written as a tensor product of quadratic forms as:

:$langle\; langle\; a\_1,\; a\_2,\; ...\; ,\; a\_n\; angle\; angle\; cong\; langle\; 1,\; -a\_1\; angle\; otimes\; langle\; 1,\; -a\_2\; angle\; otimes\; ...\; otimes\; langle\; 1,\; -a\_n\; angle$,

For "a_i" elements of the field "F". An n-Pfister form may also be constructed inductively from an "n-1"-Pfister form "q" and an "a" in "F", as $q\; oplus\; (-a)q$.

So all 1-Pfister forms and 2-Pfister forms look like:

:$langle\; 1,\; -a\; angle\; cong\; x^2\; -\; ay^2$.:$langle\; 1,\; -a,\; -b,\; ab\; angle\; cong\; x^2\; -\; ay^2\; -bz^2\; +abw^2$.

"n"-Pfister forms for n ≤ 3 are norm forms of composition algebras. In fact, in this case, two "n"-Pfister forms are isometric if and only if the corresponding composition algebras are isomorphic.

References

* | year=2004 | volume=67, Ch. 10