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The 29 pages that link to Field extension, each with the reason it gives.

• Algebraic numberNarrower topic: Adjoining an algebraic number creates a finite extension of the rational numbers.

• Field (mathematics)Related: Extensions enlarge a field while preserving its arithmetic operations.

• Galois groupNarrower topic: A Galois group is defined relative to this inclusion.

• Évariste GaloisRelated: Adjoining polynomial roots creates the extensions whose symmetries Galois theory studies.

• Cyclotomic fieldNarrower topic: Adjoining a root of unity creates an extension of the rational numbers.

• Galois theoryNarrower topic: The theory studies how automorphisms of extensions encode their intermediate fields.

• Minimal polynomialNarrower topic: The base field and algebraic element sit inside this larger field structure.

• Splitting fieldNarrower topic: A splitting field is a particular extension built by adjoining a polynomial’s roots.

• FieldRelated: Extensions describe how fields acquire solutions to additional polynomial equations.

• Angle trisectionRelated: Constructible lengths arise through restricted sequences of quadratic field extensions.

• Doubling the cubeRelated: Compass-and-straightedge constructions produce numbers through extensions of degree a power of two.

• Lindemann–Weierstrass theoremRelated: Algebraic numbers form a field extension of the rationals, where the theorem's coefficients lie.

• Squaring the circleRelated: Constructible lengths arise through field extensions of degree that is a power of two.

• Constructible numberNarrower topic: Each construction step enlarges the current coordinate field by at most a quadratic extension.

• Straightedge and compass constructionRelated: Constructible coordinates arise through successive quadratic field extensions.

• Algebraically closed fieldRelated: Roots missing from a field can be added by passing to an extension.

• Field automorphismRelated: Automorphisms often become significant when they fix a base field inside an extension.

• Fundamental theorem of Galois theoryNarrower topic: The theorem applies to extensions, with a finite Galois extension as its central setting.

• Solvability by radicalsNarrower topic: Adjoining radicals creates a tower of field extensions in the algebraic criterion.

• Algebraic number fieldNarrower topic: A number field is specifically a finite extension of the rational field.

• Hyperreal numberRelated: The hyperreals extend the real field without changing its embedded elements.

• Normal extensionNarrower topic: Normality is a property of one field viewed as an extension of another.

• Kronecker–Weber theoremNarrower topic: The theorem classifies a particular family of finite extensions of the rationals.

• Primitive element theoremNarrower topic: The theorem concerns finite extensions of this kind.

• Jordan–Chevalley decompositionRelated: The meaning of semisimple can depend on whether diagonalizability is tested over the base field or an extension.

• Noether normalization lemmaRelated: The lemma is formulated over a field, whose coefficients enable the required changes of generators.

• Skolem–Noether theoremRelated: The base field and its extensions set the scalars for the algebras and embeddings.

• Dieudonné's theoremRelated: Different fields can have different automorphisms, changing which semilinear symmetries are possible.

• Lüroth's theoremNarrower topic: The theorem concerns the structure of subfields inside a field extension.