Splitting field

The smallest field extension of a base field in which a given polynomial factors into linear factors. It is unique up to isomorphism fixing the base field.

Field extension: A field containing a smaller field as a subfield. A splitting field is a particular extension built by adjoining a polynomial’s roots.
Adjoining a root: The construction of a larger field by adding a root of a polynomial. Repeated root adjunctions produce a field where the polynomial splits.
Galois theory: The study of field extensions through their groups of symmetries. Splitting fields connect polynomial roots to the groups that govern their algebraic relations.
Normal extension: An algebraic extension in which every irreducible polynomial with a root in the extension splits there. A splitting field is normal over its base field when the polynomial is irreducible.
Polynomial: An expression formed from coefficients and nonnegative integer powers of a variable. The polynomial determines which roots the splitting field must contain.
Minimal polynomial: The monic irreducible polynomial of least degree satisfied by an algebraic element. Its roots determine the conjugates that must appear alongside an adjoined root.
Solvability by radicals: The expression of polynomial roots using arithmetic operations and nested radicals. Galois groups of splitting fields determine whether a polynomial is solvable by radicals.
Separable extension: An algebraic extension in which minimal polynomials have distinct roots. Separability determines whether the splitting field’s degree matches its automorphism count.
Root of a polynomial: An element that makes a polynomial evaluate to zero. The extension must contain every root, including roots absent from the base field.
Tower law: The degree formula relating a tower of finite field extensions to its component degrees. It computes the degree as roots are adjoined through intermediate fields.

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