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getBaseField -- returns the base field of a stable or unstable Grothendieck-Witt class

Description

Given a GrothendieckWittClass or UnstableGrothendieckWittClass object beta, this method returns the base field of the Grothendieck-Witt class if the Grothendieck-Witt class is defined over a field.

i1 : R = QQ[x]/(x^2 + 1);
i2 : beta = makeGWClass matrix(R, {{1,2},{2,x}});
i3 : getBaseField beta

o3 = R[]

o3 : PolynomialRing

Caveat

This method will return the base field of the (unstable) Grothendieck-Witt class if it is defined over a finite separable extension of either the rational numbers or a finite field of characteristic not 2. The method verifies that the zero ideal of the getAlgebra of the (unstable) Grothendieck-Witt class is prime using isPrime, which only works for ideals in polynomial rings or quotients of polynomial rings over a prime field. In particular, this method may fail even when the (unstable) Grothendieck-Witt class is indeed defined over a field. In these cases, the user can apply toField to the algebra of the (unstable) Grothendieck-Witt class to obtain the base field.

See also

Ways to use getBaseField:

  • getBaseField(GrothendieckWittClass)
  • getBaseField(UnstableGrothendieckWittClass)

For the programmer

The object getBaseField is a method function.


The source of this document is in A1BrouwerDegrees/Documentation/UnstableGrothendieckWittClassesDoc.m2:188:0.