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toField(Ring) -- declare that a ring is a field

Synopsis

Description

i1 : A = QQ[i]/(i^2+1);
i2 : toField A

o2 = A

o2 : QuotientRing
i3 : B = A[x,y,z]

o3 = B

o3 : PolynomialRing
i4 : I = ideal(i*x^2-y-i, i*y^2-z-i)

               2             2
o4 = ideal (i*x  - y - i, i*y  - z - i)

o4 : Ideal of B
i5 : gens gb I

o5 = | y2+zi-1 x2+yi-1 |

             1       2
o5 : Matrix B  <--- B
The declaration is accomplished by setting R.isField to be true, and, in case the ring is a ring handled by the engine, informing the engine. Polynomial rings over rings declared to be fields support Groebner basis operations.

If the engine eventually discovers that some nonzero element of R is not a unit, an error will be signalled. The user may then use getNonUnit to obtain a non-invertible element of R. If a ring is probably a field, it can be used as a field until a contradiction is found, and this may be a good way of discovering whether a ring is a field.

See also