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

Synopsis

• Usage:

  PfisterForm(k,a)

  PfisterForm(k,L)

• Inputs:
  • k, a ring, a field
  • a, a ring element, any element $a\in k$
  • L, a sequence, of elements $L = (a_1,\ldots,a_n)$ with $a_i \in k$
• Outputs:
  • a Grothendieck Witt Class, 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