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The smallest field extension over which a specified family of polynomials splits into linear factors.
wp-splitting-fieldnlab-splitting-fieldmilne-fields-galois-theoryEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Choose a polynomial or family over the base field and a field generated by all of its roots. The resulting splitting field is unique up to base-field isomorphism, not as a literally canonical subfield without an ambient algebraic closure.
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.
Supply the splitting field generated by the roots of the polynomial under study.
Feed several structures into a construction junction and impose compatibility between them.
milne-fields-galois-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
A splitting field of a family of polynomials over the base field is algebraic and normal over that field; forgetting the chosen polynomials leaves the normal extension.
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