Splitting fields: put every conjugate in the same arithmetic universe

A splitting field of a nonconstant polynomial f over F is generated over F by all of its roots, in a field where f factors into linear factors. The generation requirement excludes unnecessary extra elements. To construct one, choose an irreducible factor, adjoin one of its roots through a polynomial quotient, factor out the new linear factor, and repeat on the remaining polynomial. Each adjoining step is finite, so the tower law proves the resulting extension is finite.

Repeated roots do not require repeated adjoining steps. Once a root is available, all of its repeated copies are already present; multiplicity is a property of the factorization, not a demand for different elements. A splitting field is unique up to an isomorphism fixing F, although that isomorphism need not be unique. Different choices of a first root can therefore produce differently presented but equivalent fields. This is why an algebraic construction should not depend on a favorite decimal approximation.

Let α be the real cube root of 3 and let ζ be a nonreal cube root of unity. The roots are α, αζ, and αζ². The field Q(α) is contained in the real numbers, so it cannot contain the last two roots. Adjoining ζ supplies all three, giving Q(α, ζ). Conversely, any splitting field containing α and αζ also contains their ratio ζ, so no smaller root-generated field suffices.

Eisenstein at 3 makes X³ − 3 irreducible over Q, hence [Q(α):Q] is three. The element ζ satisfies X² + X + 1. This polynomial has no root in the real field Q(α), so it remains irreducible there and contributes degree two. The splitting field therefore has degree six. The polynomial has degree three, but its splitting field needs six coordinates because adjoining one root does not yet provide every conjugate.

An algebraically closed field is one in which every nonconstant polynomial has a root, and therefore splits by repeated division. An algebraic closure of F is an algebraically closed extension whose elements are all algebraic over F. The second condition matters. The complex numbers are algebraically closed and contain Q, but they are not an algebraic closure of Q because they contain transcendental numbers. The complex numbers are instead an algebraic closure of the real numbers, with degree two.

Inside C, the algebraic numbers form an algebraic closure of Q. This field is infinite-dimensional over Q, even though every individual element has a finite-degree minimal polynomial. More generally, algebraic closures exist and are unique up to base-field isomorphism; a general proof uses a maximality or extension argument. This existence result provides a shared destination for roots and embeddings without asserting that every polynomial has an explicit radical formula. Algebraic existence and a chosen method of calculation are separate questions.