Algebraically closed field

An algebraically closed field is a field F with the property that every non-constant polynomial with coefficients in F has at least one root in F. Equivalently, every non-constant polynomial in F[x] splits completely into linear factors over F. The most familiar example is the field ℂ of complex numbers, whose algebraic closedness is the content of the fundamental theorem of algebra. Algebraically closed fields occupy a central position in abstract algebra: they are the natural setting in which polynomial equations can always be solved, they serve as the base fields of classical algebraic geometry, and they form one of the fundamental classes of structures in mathematical logic and model theory. For a field K, an algebraically closed algebraic extension of K is called an algebraic closure of K, usually denoted K̄ or K^alg.

Definition and Basic Characterizations

Let F be a field. The following conditions are equivalent, and a field satisfying them is said to be algebraically closed:

1. Every non-constant polynomial f ∈ F[x] has a root in F.
2. Every polynomial in F[x] of positive degree factors as a product of linear polynomials over F.
3. The only irreducible polynomials in F[x] are those of degree one.
4. F has no proper algebraic field extensions; that is, whenever F ⊆ E is an algebraic extension, then E = F.
5. Every polynomial x^n − a with a ∈ F and n ≥ 1 has a solution in F, and every element of F admits all roots of unity; equivalently, F is "radically closed" and contains all roots of unity.
6. Every square matrix over F has an eigenvalue in F (equivalently, every characteristic polynomial splits over F).

The equivalence of (1) and (2) follows by induction on the degree: if a polynomial of degree n ≥ 2 has a root a, then it factors as (x − a)g(x) with deg g = n − 1, and the induction proceeds. Condition (3) is immediate from (2), and (4) expresses the maximality of F among algebraic extensions: any root of an irreducible polynomial f ∈ F[x] generates an algebraic extension of F, which must be trivial if F is already algebraically closed.

Two immediate consequences are worth noting. First, an algebraically closed field is necessarily infinite, since a finite field F~q~ cannot be algebraically closed: the polynomial x^(q+1) − 1 has q + 1 distinct roots in any splitting field (its derivative does not vanish at any root), yet the multiplicative group of F~q~ has only q − 1 elements, so the polynomial cannot split in F~q~. Second, an algebraically closed field of characteristic p > 0 is perfect, because the Frobenius map x ↦ x^p is surjective (every element has a p-th root).

Examples and Non-examples

Examples.

• The field ℂ of complex numbers, by the fundamental theorem of algebra.
• The field ℚ̄ of algebraic numbers, i.e., the set of complex numbers algebraic over ℚ. It is countable, a striking counterpoint to the uncountability of ℂ.
• The algebraic closure 𝔽̄~p~ = ⋃~n~ 𝔽~p^n~ of the finite field 𝔽~p~, a countable union of finite fields that is also countable.
• The algebraic closure ℂ~p~ of ℚ̄~p~ within the completion of the p-adic numbers (see below); like ℚ̄, it is countable.
• The field of Puiseux series over an algebraically closed field of characteristic zero, by the Newton–Puiseux theorem (see below).
• More generally, for any field K and any set of indeterminates, the algebraic closure of K is algebraically closed; inside ℂ, for instance, the relative algebraic closure of ℚ(S) for any set S of complex numbers is an algebraically closed subfield.

Non-examples.

• ℝ is not algebraically closed, since x² + 1 has no real root. More structurally, in an ordered field all squares are non-negative, so −1 can never be a square; no algebraically closed field can be ordered.
• ℚ is not algebraically closed (x² − 2 has no rational root), and neither is any number field.
• The rational function field ℝ(t) (or K(t) for any K) is not algebraically closed.
• No finite field is algebraically closed, by the degree-counting argument above.

Historical Development

The origins of the concept lie in the fundamental theorem of algebra, which asserts that ℂ is algebraically closed. Early formulations appear in the work of Albert Girard (1629) and René Descartes, who recognized that a polynomial of degree n should have n roots once complex quantities are admitted. Jean le Rond d'Alembert attempted a proof in 1746, and further efforts were made by Euler and Lagrange, though all contained