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
• Generalized decomposition columns have corresponding block support 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
• An algebraic closure of a field Definition
• Classical affine algebraic sets, including the empty boundaries Definition
• Classical algebraic prevarieties, regular maps, and varieties Definition
• A p-section with no local block inducing to the chosen global block Example
• The Second Main Theorem at u=1 is block-diagonal decomposition Example
• FALSE: the real numbers are algebraically closed False statement
• Additive characters are exactly one-dimensional complex representation characters Lemma
• Classical k-points give closed points over an algebraically closed field Lemma
• Finite-type field extensions with zero Ω 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
• The one-step root condition makes an algebraic extension of a perfect field algebraically closed Lemma
• To prove the fundamental theorem of algebra, it suffices to split every real polynomial over ℂ Lemma
• A field is algebraically closed exactly when every nonconstant polynomial splits, equivalently when it has no nontrivial finite extension Proposition
• Assuming Choice, a base-field embedding extends across every algebraic extension Theorem
• Brauer's Second Main Theorem Theorem
• F-embeddings of F(α) into an algebraically closed field correspond to the distinct roots of m_α Theorem
• Green indecomposability for index-p integral induction Theorem
• The binary Sylvester resultant detects a common geometric projective root Theorem

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)