KnowraAlgebraic closureLinked fromLinked fromThe 14 pages that link to Algebraic closure, each with the reason it gives.All 14Broader topic 1Related 6Narrower topic 5Compared with 2Fundamental theorem of algebraNarrower topic: The theorem says the complex field has precisely this root-existence property.Algebraic numberNarrower topic: The algebraic numbers form the algebraic closure of the rational numbers inside the complex numbers.Field extensionNarrower topic: It provides a common setting in which polynomial roots and their extensions can be studied.Galois groupNarrower topic: Automorphisms of an algebraic closure supply the ambient symmetries behind finite Galois groups.Discriminant (polynomial)Related: Repeated-root detection is understood over an algebraic closure, even when roots are not in the coefficient field.Splitting fieldCompared with: It contains roots of every polynomial, while a splitting field contains roots of just one.FieldRelated: Every field has an algebraic closure in which all nonconstant polynomials split.Frobenius endomorphismRelated: Frobenius behavior on algebraic closures helps describe purely inseparable extensions.Algebraically closed fieldBroader topic: It is the standard construction that supplies all algebraic roots missing from a field.Solvability by radicalsRelated: It supplies a common setting in which all polynomial roots and their radical expressions can be considered.Normal extensionNarrower topic: It is the ambient setting in which normal extensions can be characterized by their embeddings.Primitive element theoremCompared with: An algebraic closure of a field is generally infinite, beyond the theorem’s finite-extension claim.Jordan–Chevalley decompositionRelated: Diagonalizability and Jordan blocks are often described after extending scalars to such a field.Gelfand–Mazur theoremRelated: The theorem depends on the complex field’s algebraic closure through spectral theory.