Cardinal Functions in Algebra
Examples of cardinal functions in algebra are:
- Index of a subgroup H of G is the number of cosets.
- Dimension of a vector space V over a field K is the cardinality of any Hamel basis of V.
- More generally, for a free module M over a ring R we define rank as the cardinality of any basis of this module.
- For a linear subspace W of a vector space V we define codimension of W (with respect to V).
- For any algebraic structure it is possible to consider the minimal cardinality of generators of the structure.
- For algebraic extensions algebraic degree and separable degree are often employed (note that the algebraic degree equals the dimension of the extension as a vector space over the smaller field).
- For non-algebraic field extensions transcendence degree is likewise used.
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