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# PfisterForm -- the Grothendieck-Witt class of a Pfister form

## Synopsis

• Usage:
PfisterForm(k,a)
PfisterForm(k,L)
• Inputs:
• k, a ring, a field
• a, , any element $a\in k$
• L, , of elements $L = (a_1,\ldots,a_n)$ with $a_i \in k$
• Outputs:
• , the Pfister form $\langle\langle a_1,\ldots,a_n\rangle\rangle \in \text{GW}(k)$

## Description

Given a sequence of elements $a_1,\ldots,a_n \in k$ we can form the Pfister form $\langle\langle a_1,\ldots,a_n\rangle\rangle$ defined to be the rank $2^n$ form defined as the product $\langle 1, -a_1\rangle \otimes \cdots \otimes \langle 1, -a_n \rangle$.

 i1 : PfisterForm(QQ,(2,6)) o1 = GrothendieckWittClass{cache => CacheTable{} } matrix => | 1 0 0 0 | | 0 -6 0 0 | | 0 0 -2 0 | | 0 0 0 12 | o1 : GrothendieckWittClass

Inputting a ring element, an integer, or a rational instead of a sequence will produce a one-fold Pfister form instead. For instance:

 i2 : PfisterForm(GF(13),-2/3) o2 = GrothendieckWittClass{cache => CacheTable{}} matrix => | 1 0 | | 0 5 | o2 : GrothendieckWittClass i3 : PfisterForm(CC,3) o3 = GrothendieckWittClass{cache => CacheTable{}} matrix => | 1 0 | | 0 -3 | o3 : GrothendieckWittClass

## Ways to use PfisterForm :

• PfisterForm(InexactFieldFamily,QQ)
• PfisterForm(InexactFieldFamily,RingElement)
• PfisterForm(InexactFieldFamily,Sequence)
• PfisterForm(InexactFieldFamily,ZZ)
• PfisterForm(Ring,QQ)
• PfisterForm(Ring,RingElement)
• PfisterForm(Ring,Sequence)
• PfisterForm(Ring,ZZ)

## For the programmer

The object PfisterForm is .