ground fields and rings
The following is a list of common uses of the ground or base field or ring in algebra. 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.
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Given a ring R with a 1, let ℤ1 be the subgroup
of R generated by 1 under addition. This is consequently a subring of R of the same characteristic
as R. Thus is it isomorphic
to ℤ/cℤ 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, ℤ1≅ℤ/pℤ and so it may be called the ground field of R.
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Given a vector space
or algebra A over a field k, then k is the ground/base field
of A.
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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.
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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 subfield
has been specified, the ground/base field then typically defaults to the prime subfield
of K. (Recall the prime subfield is the unique smallest subfield of K.)
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Given a field K and a set of field automorphisms f:K→K, the ground/base field in this context is the fixed field (http://planetmath.org/Fixed) of the automorphisms
. 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 |