Normal extension

A field extension in which every irreducible polynomial over the base field having one root in the extension splits completely there. Equivalently, it is a splitting field of a family of polynomials over the base field.

Field extension: An inclusion of fields, with the larger field regarded as a vector space over the smaller one. Normality is a property of one field viewed as an extension of another.
Field embedding: An injective homomorphism between fields that preserves addition, multiplication, and identity. Normality controls where embeddings fixing the base field can send elements.
Non-normal extension: A field extension that fails the defining root-closure condition for normality. A root may lie in the extension while other roots of its irreducible polynomial do not.
Fundamental theorem of Galois theory: The correspondence between intermediate fields and subgroups for a finite Galois extension. Normality helps make intermediate-field structure correspond cleanly to normal subgroups.
Infinite Galois theory: The study of possibly infinite algebraic Galois extensions using profinite groups and a topology on automorphisms. Normality remains central, but infinite extensions require topological versions of finite Galois correspondences.
Irreducible polynomial: A nonconstant polynomial over a field that cannot be factored into lower-degree polynomials over that field. Normality tests whether an irreducible polynomial with one extension root has all its roots there.
Minimal polynomial: The monic irreducible polynomial over a field satisfied by a given algebraic element. Its roots determine whether the element's conjugates all lie in the extension.
Purely inseparable extension: An algebraic extension in characteristic p where each element has some p-power lying in the base field. Purely inseparable extensions can be normal without being separable.
Fixed field: The subfield consisting of elements left unchanged by every automorphism in a specified group. Normal extensions connect fixed fields to subgroups of their automorphism groups.
Profinite group: A topological group that is an inverse limit of finite groups. Automorphism groups of infinite normal algebraic extensions naturally carry a profinite topology.

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