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An inclusion of fields , regarding as a vector space and algebra over .
wp-field-extensionnlab-field-extensionmilne-fields-galois-theoryEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
A field extension records an embedding of a base field into a larger field.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
milne-fields-galois-theoryEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Every element of the larger field must satisfy a nonzero polynomial over the base.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
milne-fields-galois-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
For a field extension , scalar multiplication by makes a vector space over ; its dimension is the extension degree.
Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.
milne-fields-galois-theory