An algebraically closed field: every nonconstant polynomial has a root in the field

Depends on

• Field
• Evaluation and roots of a polynomial in a commutative target ring
• Degree, leading coefficient and monic polynomial, with the zero polynomial having no degree

Used by

• Algebraic Bezout formula as a sum of local scheme lengths Corollary
• Every endomorphism of a nonzero finite-dimensional vector space over an algebraically closed field has an eigenvalue Corollary
• Every finite-dimensional endomorphism over an algebraically closed field has Jordan form Corollary
• Every finite-dimensional endomorphism over an algebraically closed field is triangularisable Corollary
• Fields of characteristic zero, finite fields, and algebraically closed fields are perfect Corollary
• General hypersurfaces give smooth complete intersections Corollary
• Generalized decomposition columns have corresponding block support Corollary
• Global functions on geometrically connected and geometrically reduced proper schemes Corollary
• Nontrivial degree-zero line bundles have no sections Corollary
• Over an algebraically closed field, every endomorphism of an irreducible representation is scalar Corollary
• Over an algebraically closed field, every maximal ideal is an evaluation ideal Corollary
• Bezout needs algebraic closure: an imaginary conic has no real point Counterexample
• Frobenius linear systems have nonreduced general members Counterexample
• An algebraic closure of a field Definition
• Classical affine algebraic sets, including the empty boundaries Definition
• Classical algebraic prevarieties, regular maps, and varieties Definition
• Linear systems of plane curves and their base loci Definition
• Linear systems, base loci, and general members Definition
• Plane projective curves and their components Definition
• Tangent cone and tangent lines at a point Definition
• A nondegenerate projective quadric Example
• A p-section with no local block inducing to the chosen global block Example
• Different singularities can share a tangent cone Example
• The finite dual of the primitive polynomial Hopf algebra Example
• The Second Main Theorem at u=1 is block-diagonal decomposition Example
• FALSE: the real numbers are algebraically closed False statement
• A dense hypersurface chart with a nonzero partial derivative Lemma
• Additive characters are exactly one-dimensional complex representation characters Lemma
• Canonical resolutions over non-algebraically-closed ground fields Lemma
• Classical k-points give closed points over an algebraically closed field Lemma
• Curves without a common component meet finitely often Lemma
• Finite-type field extensions with zero Ω Lemma
• Global intersection length is the sum of the local multiplicities Lemma
• Global length of a plane complete intersection equals the degree product Lemma
• Gram-map image, full-rank fibers, and rank-stratum dimensions Lemma
• Local block projection controls p-section character support Lemma
• Nagao error terms have zero trace on the relevant p-section Lemma
• Relative projectivity forces character vanishing off the controlling p-section Lemma
• Restriction partitions embeddings in a finite tower into extension fibres Lemma
• Scaling, specialization, and the affine and infinite charts of a binary resultant Lemma

…and 12 more results.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• M. Khovanov, Linear Algebra II notes, §6 (standard reference, not scraped)
• H. Pinkham, Linear Algebra, §12.1 (standard reference, not scraped)