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

- Usage:
    `makePfisterForm(k, a)`
    `makePfisterForm(k, L)`
- Inputs:
  - k, a ring, a field of characteristic not 2
  - a, a ring element, any element in the field
  - L, a sequence, of elements in the field $a_{i}$ where $i = 1,\dots, n$
- Outputs:
  - a Grothendieck-Witt Class, the Pfister form $\langle\langle a_{1},\ldots,a_{n}\rangle\rangle$ in the Grothendieck-Witt ring of the field

## 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 : makePfisterForm(QQ, (2,6))

    o1 = | 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 number instead of a sequence will produce a one-fold Pfister form instead. For instance:

    i2 : makePfisterForm(GF(13), -2/3)

    o2 = | 1 0 |
         | 0 5 |

    o2 : GrothendieckWittClass

    i3 : makePfisterForm(CC, 3)

    o3 = | 1 0  |
         | 0 -3 |

    o3 : GrothendieckWittClass

## Ways to use makePfisterForm:

- `makePfisterForm(InexactFieldFamily,Number)`
- `makePfisterForm(InexactFieldFamily,RingElement)`
- `makePfisterForm(InexactFieldFamily,Sequence)`
- `makePfisterForm(Ring,Number)`
- `makePfisterForm(Ring,RingElement)`
- `makePfisterForm(Ring,Sequence)`

## For the programmer

The object makePfisterForm is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/A1BrouwerDegrees/Documentation/BuildingFormsDoc.m2:77:0.