Field theory and polynomials Basic notions Concept
Field
Summary
A field is a commutative ring in which you can divide by every nonzero element. The rational numbers, the real numbers and the complex numbers are fields. The definition of a field gives the axioms.
Where this leads
108 pages build on this one.
- Unit-regular ring
- Newton polynomial
- Semifield
- Elliptic cohomology
- Algebraic set
- Cauchy matrix
- Sylvester equation
- Leavitt path algebra
- Tarski–Seidenberg theorem
- Hochster–Roberts theorem
- Hilbert polynomial
- Integral extensions of fields
- Weierstrass equation
- Gröbner basis
- Finite Desarguesian planes are Pappian
- Kummer theory
- Model theory of algebraically closed fields
- Path algebra
- Affine space as an algebra
- Noether normalization lemma
- Central polynomial
- Fusion category
- Bernstein–Gelfand–Ponomarev reflection functors
- Residue field
- Pascal's theorem
- Central simple algebra
- Frobenius algebra
- Artinian reduced rings are semisimple
- Combinatorial Nullstellensatz
- Persistent homology
- Generalized inverse
- Quadric
- Quaternion algebra
- Generic matrix algebra
- Kurosh problem
- Pappus's hexagon theorem
- Regular chain
- Matrix pencil
- Tilting theory
- Difference algebra
- p-adic number
- Mahler's theorem
- Real closed field
- von Neumann regular ring
- Smash product
- Dodgson condensation
- Hopf algebra structure on cohomology
- abc conjecture
- Fundamental theorem of projective geometry
- Group law on an elliptic curve
- Projective space
- Deformation of algebras
- Poincaré–Birkhoff–Witt theorem
- Gabrielov's theorem of the complement
- Hilbert–Serre theorem
- Projective conic
- Multivariate division algorithm
- Weyl algebra
- Tarski's axioms
- Semialgebraic set
- Bergman's gap theorem
- Growth of algebras
- Neville's algorithm
- Hilbert's fourteenth problem
- Plücker coordinates
- Stanley–Reisner ring
- Veblen–Young theorem
- Rational point
- Coalgebra
- Pseudo algebraically closed field
- Classical group
- Incomparability theorem
- Hilbertian field
- Galois representation
- Tits alternative
- Hilbert's syzygy theorem
- Golod–Shafarevich theorem
- Local field
- Representation of a quiver
- Global field
- Elliptic curve
- Group algebra
- Newton polytope
- Universal enveloping algebra
- Morphism of finite type
- Topological modular forms
- Divided differences
- Ring of differential operators
- Gerstenhaber algebra
- ℓ-adic cohomology
- Barycentric interpolation formula
- Elimination theory
- Pappus configuration
- Exceptional collection
- Fundamental theorem of affine geometry
- Algebraic number field
- Hilbert scheme
- Toric ideal
- Zariski's lemma
- Buchberger's algorithm
- Koszul algebra
- Surreal number
- Koszul duality
- Gelfand–Kirillov dimension
- Monomial ideal
- Lagrange basis polynomials
- Drazin inverse
- Mason–Stothers theorem
Further reading
- Web https://en.wikipedia.org/wiki/Field_(mathematics), lead section
About this page
| Tag | 04C5ZBG1WT |
|---|---|
| URI | m2://04C5ZBG1WT |
| Kind | concept |
| Writing | W1 stated |
| MSC | 12E99 |
| Specializes | Division ring |
| Same as |
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What this page needs for W2
- edge instance_of in: at least two examples (0 of 2)
- edge not_instance_of in: at least one non-example (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.
What links here
Special cases
- Ordered field
- Algebraically closed field
- Finite field
- Global field
- Separably closed field
- Topological field
- Differential field
- Local field
- Algebraic number field
- Residue field
Definitions
Used by
- Newton polynomial
- Semifield
- Algebraic set
- Cauchy matrix
- Sylvester equation
- Leavitt path algebra
- Hochster–Roberts theorem
- Hilbert polynomial
- Integral extensions of fields
- Weierstrass equation
- Gröbner basis
- Kummer theory
- Model theory of algebraically closed fields
- Path algebra
- Affine space as an algebra
- Noether normalization lemma
- Central polynomial
- Fusion category
- Bernstein–Gelfand–Ponomarev reflection functors
- Residue field
- Pascal's theorem
- Central simple algebra
- Frobenius algebra
- Artinian reduced rings are semisimple
- Combinatorial Nullstellensatz
- Persistent homology
- Generalized inverse
- Quadric
- Quaternion algebra
- Generic matrix algebra
- Kurosh problem
- Pappus's hexagon theorem
- Regular chain
- Matrix pencil
- Tilting theory
- Difference algebra
- p-adic number
- Real closed field
- von Neumann regular ring
- Smash product
- Dodgson condensation
- Hopf algebra structure on cohomology
- Fundamental theorem of projective geometry
- Group law on an elliptic curve
- Projective space
- Deformation of algebras
- Poincaré–Birkhoff–Witt theorem
- Hilbert–Serre theorem
- Projective conic
- Multivariate division algorithm
- Weyl algebra
- Bergman's gap theorem
- Growth of algebras
- Neville's algorithm
- Hilbert's fourteenth problem
- Plücker coordinates
- Stanley–Reisner ring
- Rational point
- Coalgebra
- Pseudo algebraically closed field
- Classical group
- Hilbertian field
- Galois representation
- Tits alternative
- Hilbert's syzygy theorem
- Golod–Shafarevich theorem
- Local field
- Representation of a quiver
- Global field
- Elliptic curve
- Group algebra
- Newton polytope
- Universal enveloping algebra
- Divided differences
- Ring of differential operators
- Gerstenhaber algebra
- Barycentric interpolation formula
- Elimination theory
- Exceptional collection
- Fundamental theorem of affine geometry
- Algebraic number field
- Toric ideal
- Zariski's lemma
- Buchberger's algorithm
- Koszul algebra
- Koszul duality
- Gelfand–Kirillov dimension
- Monomial ideal
- Lagrange basis polynomials
- Drazin inverse
- Mason–Stothers theorem