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 A1BrouwerDegrees/Documentation/BuildingFormsDoc.m2:77:0.