Primitive element theorem

The primitive element theorem states that every finite separable field extension is generated by one element. In particular, every finite extension of a perfect field has a primitive element.

Primitive element: An element that generates a field extension over its base field. The theorem asserts that such an element exists for every finite separable extension.
Field extension: An inclusion of fields, viewed through the larger field’s structure over the smaller one. The theorem concerns finite extensions of this kind.
Simple extension: A field extension generated over its base field by one element. The theorem identifies every finite separable extension as a simple extension.
Purely inseparable extension: An algebraic extension in characteristic p in which each element has some p-power lying in the base field. Such extensions can fail to be generated by one element, unlike finite separable extensions.
Separability: A property of an algebraic extension in which minimal polynomials have distinct roots in a splitting field. Separability is the theorem’s key hypothesis.
Finite extension: A field extension that has finite dimension as a vector space over its base field. Finiteness is part of the theorem’s main assumption.
Galois extension: A finite field extension that is both normal and separable. Its separability guarantees a primitive element, enabling a one-generator description.
Imperfect field: A field that has an algebraic extension that is not separable. Finite extensions over an imperfect field need not meet the theorem’s hypothesis.
Minimal polynomial: The monic polynomial of least positive degree over a field that has a given algebraic element as a root. A primitive element encodes the extension through its minimal polynomial.
Algebraic extension: A field extension in which every element satisfies a nonzero polynomial over the base field. Every finite field extension is algebraic, providing the setting for minimal polynomials.

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