KnowraCyclotomic fieldLinked fromLinked fromThe 19 pages that link to Cyclotomic field, each with the reason it gives.All 19Broader topic 12Related 6Narrower topic 1Algebraic number theoryBroader topic: Cyclotomic fields connect ideal arithmetic with roots of unity and classical Diophantine equations.Field extensionBroader topic: These explicit extensions encode roots of unity and their arithmetic symmetries.Galois groupBroader topic: Over the rationals, its Galois group is described by invertible residue classes modulo the root's order.Root of unityBroader topic: Adjoining these numbers produces central examples in algebraic number theory.Cyclotomic polynomialBroader topic: A primitive nth root generates a field whose minimal polynomial is Φₙ(x).Galois theoryBroader topic: Cyclotomic extensions give explicit, structured examples of Galois groups and radical solutions.Splitting fieldBroader topic: It is the splitting field over the rationals of a cyclotomic polynomial.Class field theoryRelated: Cyclotomic fields provide concrete examples of abelian extensions and reciprocity.Ernst KummerRelated: Kummer studied these fields while attacking Fermat’s last theorem.Algebraic integerBroader topic: Roots of unity are algebraic integers, central to cyclotomic arithmetic.Field automorphismRelated: Its automorphisms act explicitly by sending a primitive root of unity to its coprime powers.Fundamental theorem of Galois theoryRelated: Its Galois group and intermediate fields can be read through the theorem’s subgroup correspondence.Algebraic number fieldBroader topic: It is a concrete family of number fields with especially rich Galois structure.Dirichlet's unit theoremRelated: Its real and complex embeddings give explicit examples of the rank formula.Normal extensionBroader topic: Cyclotomic fields over the rationals are standard finite Galois, hence normal, extensions.Gauss–Wantzel theoremRelated: Its degree encodes the algebraic complexity of the polygon's vertices.Kronecker–Weber theoremBroader topic: The theorem places every finite abelian extension inside one of these fields.Primitive element theoremBroader topic: Cyclotomic extensions provide concrete examples of finite extensions described by generators.Kummer–Vandiver conjectureNarrower topic: The conjecture concerns the cyclotomic field generated by a primitive pth root of unity.
KnowraCyclotomic fieldLinked fromLinked fromThe 19 pages that link to Cyclotomic field, each with the reason it gives.All 19Broader topic 12Related 6Narrower topic 1Algebraic number theoryBroader topic: Cyclotomic fields connect ideal arithmetic with roots of unity and classical Diophantine equations.Field extensionBroader topic: These explicit extensions encode roots of unity and their arithmetic symmetries.Galois groupBroader topic: Over the rationals, its Galois group is described by invertible residue classes modulo the root's order.Root of unityBroader topic: Adjoining these numbers produces central examples in algebraic number theory.Cyclotomic polynomialBroader topic: A primitive nth root generates a field whose minimal polynomial is Φₙ(x).Galois theoryBroader topic: Cyclotomic extensions give explicit, structured examples of Galois groups and radical solutions.Splitting fieldBroader topic: It is the splitting field over the rationals of a cyclotomic polynomial.Class field theoryRelated: Cyclotomic fields provide concrete examples of abelian extensions and reciprocity.Ernst KummerRelated: Kummer studied these fields while attacking Fermat’s last theorem.Algebraic integerBroader topic: Roots of unity are algebraic integers, central to cyclotomic arithmetic.Field automorphismRelated: Its automorphisms act explicitly by sending a primitive root of unity to its coprime powers.Fundamental theorem of Galois theoryRelated: Its Galois group and intermediate fields can be read through the theorem’s subgroup correspondence.Algebraic number fieldBroader topic: It is a concrete family of number fields with especially rich Galois structure.Dirichlet's unit theoremRelated: Its real and complex embeddings give explicit examples of the rank formula.Normal extensionBroader topic: Cyclotomic fields over the rationals are standard finite Galois, hence normal, extensions.Gauss–Wantzel theoremRelated: Its degree encodes the algebraic complexity of the polygon's vertices.Kronecker–Weber theoremBroader topic: The theorem places every finite abelian extension inside one of these fields.Primitive element theoremBroader topic: Cyclotomic extensions provide concrete examples of finite extensions described by generators.Kummer–Vandiver conjectureNarrower topic: The conjecture concerns the cyclotomic field generated by a primitive pth root of unity.