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A commutative ring in which every nonzero element has a multiplicative inverse.
The inverse of each nonzero element is unique. The convention excludes the zero ring.
wp-fieldnlab-fieldstacks-algebra-textEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Require and every nonzero element to be invertible; inverse elements are unique.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-fieldnlab-fieldstacks-algebra-textThis is an authored directed relation from the source endpoint to the target endpoint.
Elements fixed by a group of field automorphisms are closed under the field operations and form a subfield.
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-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
The quotient of a local ring by its unique maximal ideal is a field.
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.
stacks-algebra-textwp-fieldnlab-fieldEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Require every nonconstant polynomial to have a root.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-algebraically-closednlab-algebraically-closedstacks-algebra-textThis is an authored directed relation from the source endpoint to the target endpoint.
Retain the commutative-ring operations and forget the nonzero-invertibility axiom.
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.
wp-fieldnlab-fieldstacks-algebra-textThis is an authored directed relation from the source endpoint to the target endpoint.
Every field is in particular a division ring; retain the same operations while viewing commutativity as extra.
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.
wp-fieldnlab-fieldstacks-algebra-textThis is an authored directed relation from the source endpoint to the target endpoint.
Field automorphisms form a group under composition.
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-theoryThis 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-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Restrict to fields with finitely many elements.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-finite-fieldmilne-algebraic-number-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Require a finite extension of or .
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-global-fieldnlab-global-fieldmilne-algebraic-number-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Use the field structure on the common carrier.
Feed several structures into a construction junction and impose compatibility between them.
wp-ordered-fieldgierz-orderThis is an authored directed relation from the source endpoint to the target endpoint.
Supply field operations.
Feed several structures into a construction junction and impose compatibility between them.
wp-top-fieldtopology-without-tearsThis is an authored directed relation from the source endpoint to the target endpoint.
Supply the Field structure as one jointly required input.
Feed several structures into a construction junction and impose compatibility between them.
wp-varietynlab-varietystacks-schemes-textThis is an authored directed relation from the source endpoint to the target endpoint.
Supply the scalar field.
Feed several structures into a construction junction and impose compatibility between them.
wp-vectorstacks-algebra-textThis is an authored directed relation from the source endpoint to the target endpoint.
Require characteristic zero and finite dimension over the canonically embedded prime field .
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
nlab-number-fieldmilne-algebraic-number-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Require every algebraic extension to be separable.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-perfect-fieldstacks-algebra-textThis is an authored directed relation from the source endpoint to the target endpoint.
The rational numbers form the characteristic-zero prime field, equipped here with their canonical order.
The target is a member or subtype of the broader source class.
wp-rational-numbernlab-rational-numberThis is an authored directed relation from the source endpoint to the target endpoint.
Equip the field with a valuation or absolute value.
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.
wp-valued-fieldmilne-algebraic-number-theory