Field (mathematics) - Some First Theorems

Some First Theorems

• Every finite subgroup of the multiplicative group F× is cyclic. This applies in particular to F_q×, it is cyclic of order q − 1. In the introductory example, a generator of F₄× is the element A.

• From the point of view of algebraic geometry, fields are points, because the spectrum Spec F has only one point, corresponding to the 0-ideal. This entails that a commutative ring is a field if and only if it has no ideals except {0} and itself. Equivalently, an integral domain is field if and only if its Krull dimension is 0.

• Isomorphism extension theorem