ground fields and rings

The following is a list of common uses of the ground or base field or ring in algebraPlanetmathPlanetmath. These are endowed with based on their context so the following list may be or may not apply uniformly.

One commonality is generally found for the use of ground ring or field: the result is a unitial subring of the original. Outside of this requirement, the constraints are specific to context.

  • Given a ring R with a 1, let 1 be the subgroupMathworldPlanetmathPlanetmath of R generated by 1 under addition. This is consequently a subring of R of the same characteristicPlanetmathPlanetmath as R. Thus is it isomorphicPlanetmathPlanetmathPlanetmath to /c where c is the characteristic of R. This is the smallest unital subring of R and so rightfully may be called the ground or base ring of R.

    When the characteristic of R is prime, 1/p and so it may be called the ground field of R.

  • Given a vector spaceMathworldPlanetmath or algebra A over a field k, then k is the ground/base fieldMathworldPlanetmath of A.

  • Given a set of matrices Mn(R), the ground ring is commonly the ring R, and if required as a subring of Mn(R) then it is taken as the set of all scalar matrices.

  • Given a field extension K/k over a field k, then k is the ground field of K in this context. For a general field where no specific subfieldMathworldPlanetmath has been specified, the ground/base field then typically defaults to the prime subfieldMathworldPlanetmath of K. (Recall the prime subfield is the unique smallest subfield of K.)

  • Given a field K and a set of field automorphismsf:KK,  the ground/base field in this context is the fixed field ( of the automorphismsPlanetmathPlanetmathPlanetmathPlanetmath. That is, the largest subfield of K which is pointwise fixed by each f. Since a field automorphism must fix the prime subfield, this definition always produces a field containing the prime subfield.

Title ground fields and rings
Canonical name GroundFieldsAndRings
Date of creation 2013-03-22 15:54:22
Last modified on 2013-03-22 15:54:22
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 15
Author Algeboy (12884)
Entry type Definition
Classification msc 08A30
Related topic ExtensionField
Related topic FieldAdjunction
Related topic RingAdjunction
Defines ground field
Defines base field
Defines ground ring
Defines base ring