Field extension

A field extension is a field containing a smaller field as a subfield, with the same addition and multiplication on that subfield. It allows elements outside the smaller field to be studied algebraically.

Field: A set with addition, subtraction, multiplication, and division by nonzero elements satisfying the field axioms. An extension is itself a field, while its smaller field sits inside it.
Degree of a field extension: The dimension of a larger field as a vector space over its subfield. Degree gives a precise measure of the size of a finite extension.
Algebraic closure: An algebraically closed algebraic extension of a field, containing roots of every nonconstant polynomial over it. It provides a common setting in which polynomial roots and their extensions can be studied.
Separable extension: An algebraic extension in which every element has a minimal polynomial with distinct roots in an algebraic closure. Separable extensions have no repeated-root obstruction in their defining polynomials.
Subfield: A subset of a field that is itself a field under the inherited addition and multiplication. The smaller field in an extension must be a subfield of the larger one.
Basis (linear algebra): A linearly independent set of vectors that spans a vector space. A basis over the base field counts the independent directions in the extension.
Splitting field: The smallest field extension in which a given polynomial factors completely into linear factors. It is the minimal extension needed to contain all roots of a polynomial.
Inseparable extension: An algebraic extension containing an element whose minimal polynomial has repeated roots in an algebraic closure. Inseparability marks a distinct kind of algebraic extension, possible over fields of positive characteristic.
Field generated by a set: The smallest subfield containing a given field and a specified set of elements. Adjoining elements constructs extensions as fields generated over a base field.
Tower law: The rule that degrees multiply in a tower of finite field extensions. It lets degrees of nested extensions be computed from their steps.

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