Field theory and polynomials Algebraic closure Concept
Algebraically closed field
Summary
An algebraically closed field is a field in which every nonconstant polynomial has a root. The complex numbers are the main example. The definition of an algebraically closed field gives equivalent conditions.
Where this leads
36 pages build on this one.
- Enriques–Kodaira classification
- Minimal model program
- Ideal of an algebraic set
- Model theory of algebraically closed fields
- Algebraic surface
- Differentially closed field
- Torelli theorem
- Schottky problem
- Baily–Borel compactification
- Algebraic variety
- Hilbert's Nullstellensatz
- Coordinate ring
- Brauer character
- Positivstellensatz
- Rational surface
- Jacobian criterion
- Picard–Vessiot extension
- Genus of a curve
- Projective variety
- Cubic surface
- Ax–Grothendieck theorem
- j-invariant of an elliptic curve
- Kovacic's algorithm
- Genus–degree formula
- Pseudo algebraically closed field
- Clifford's theorem on special divisors
- Lefschetz principle
- Elimination theory
- Twenty-seven lines on a cubic surface
- Hyperelliptic curve
- Lüroth problem
- Castelnuovo's contraction theorem
- Riemann–Roch theorem
- Affine variety
- Cone theorem
- Weak Nullstellensatz
Further reading
About this page
| Tag | 0PF6TM8TWT |
|---|---|
| URI | m2://0PF6TM8TWT |
| Kind | concept |
| Writing | W1 stated |
| MSC | 12F05 |
| Specializes | Field |
| Same as |
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What this page needs for W2
- edge instance_of in: at least two examples (1 of 2)
- edge not_instance_of in: at least one non-example (0 of 1)
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What links here
Definitions
Special cases
Used by
- Ideal of an algebraic set
- Model theory of algebraically closed fields
- Algebraic surface
- Differentially closed field
- Torelli theorem
- Algebraic variety
- Hilbert's Nullstellensatz
- Brauer character
- Rational surface
- Jacobian criterion
- Picard–Vessiot extension
- Genus of a curve
- Projective variety
- Ax–Grothendieck theorem
- j-invariant of an elliptic curve
- Genus–degree formula
- Pseudo algebraically closed field
- Lefschetz principle
- Elimination theory
- Twenty-seven lines on a cubic surface
- Hyperelliptic curve
- Lüroth problem
- Castelnuovo's contraction theorem
- Riemann–Roch theorem
- Affine variety
- Cone theorem
- Weak Nullstellensatz