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makeHyperbolicForm -- the Grothendieck-Witt class of a hyperbolic form

Usage:

makeHyperbolicForm(k)

makeHyperbolicForm(k, n)
Inputs:
- k, a ring, a field or finite étale algebra over a field
- n, an integer, an even number, giving an optional rank $n$ for a totally hyperbolic form
Outputs:
- a Grothendieck-Witt Class, the hyperbolic form $\mathbb{H} = \langle 1, -1\rangle$ in the Grothendieck-Witt ring or the totally hyperbolic form $\left(\frac{n}{2}\right)\mathbb{H}$ if an optional rank $n$ is specified

Description

By default outputs the rank two hyperbolic form over the input algebra.

i1 : makeHyperbolicForm(GF(7))

o1 = | 1 0  |
     | 0 -1 |

o1 : GrothendieckWittClass

Specifying a rank yields a copy of sums of the rank two hyperbolic form. Only even rank inputs are accepted.

i2 : makeHyperbolicForm(RR, 4)

o2 = | 1 0  0 0  |
     | 0 -1 0 0  |
     | 0 0  1 0  |
     | 0 0  0 -1 |

o2 : GrothendieckWittClass

See also

isAnisotropic -- determines whether a Grothendieck-Witt class is anisotropic
getSumDecomposition -- produces a simplified diagonal representative of a Grothendieck-Witt class or unstable Grothendieck-Witt class
getSumDecompositionString -- produces a simplified string representation of a Grothendieck-Witt class or unstable Grothendieck-Witt class

Ways to use makeHyperbolicForm:

makeHyperbolicForm(InexactFieldFamily)
makeHyperbolicForm(InexactFieldFamily,ZZ)
makeHyperbolicForm(Ring)
makeHyperbolicForm(Ring,ZZ)

For the programmer

The object makeHyperbolicForm is a method function.

The source of this document is in A1BrouwerDegrees/Documentation/BuildingFormsDoc.m2:112:0.