Algebraic closure

In mathematics, specifically in abstract algebra, an algebraic closure of a field $F$ is an algebraic extension of $F$ that is algebraically closed. It represents the maximal algebraic extension of a given field, containing all roots of all non-constant polynomials with coefficients in the base field, and serves as a foundational concept in Galois theory, algebraic geometry, and number theory.

Definition and Basic Properties

Let $F$ be a field. A field extension $\overline{F}$ of $F$ is called an algebraic closure of $F$ if it satisfies two conditions:

1. $\overline{F}$ is an algebraic extension of $F$, meaning every element of $\overline{F}$ is a root of some non-zero polynomial with coefficients in $F$.
2. $\overline{F}$ is algebraically closed, meaning every non-constant polynomial with coefficients in $\overline{F}$ has at least one root in $\overline{F}$.

From these conditions, it follows that $\overline{F}$ is the "largest" algebraic extension of $F$ in the sense that any other algebraic extension of $F$ can be embedded into $\overline{F}$. Conversely, it is the "smallest" algebraically closed field containing $F$, as any algebraically closed field containing $F$ must contain a subfield isomorphic to $\overline{F}$.

Existence and Uniqueness

The existence and uniqueness of the algebraic closure for any arbitrary field were established by Ernst Steinitz in 1910.

Existence: The proof of existence typically relies on Zorn's Lemma, which is equivalent to the Axiom of Choice. One standard construction, often attributed to Emil Artin, involves iteratively adjoining roots of irreducible polynomials to the base field. By considering the polynomial ring over $F$ and constructing a maximal ideal that forces polynomials to have roots, one can build an extension where every polynomial has a root. Repeating this process transfinitely (or taking a direct limit) yields an algebraically closed field that is algebraic over $F$.

Uniqueness: Any two algebraic closures of a field $F$ are isomorphic. More precisely, if $K_1$ and $K_2$ are two algebraic closures of $F$, there exists a field isomorphism $\phi: K_1 \to K_2$ such that $\phi(x) = x$ for all $x \in F$. However, this isomorphism is generally not unique. Because the isomorphism is not canonical, mathematicians often refer to "the" algebraic closure of $F$, denoted as $\overline{F}$, with the understanding that it is unique only up to a non-unique isomorphism fixing the base field.

Examples

The nature of the algebraic closure varies significantly depending on the base field:

• Real Numbers: The algebraic closure of the field of real numbers $\mathbb{R}$ is the field of complex numbers $\mathbb{C}$. This is a finite extension of degree 2, guaranteed by the Fundamental Theorem of Algebra.
• Rational Numbers: The algebraic closure of the field of rational numbers $\mathbb{Q}$ is the field of algebraic numbers, often denoted as $\overline{\mathbb{Q}}$. This field consists of all complex numbers that are roots of non-zero polynomials with rational coefficients. It is a countably infinite extension of $\mathbb{Q}$.
• Finite Fields: For a finite field $\mathbb{F}_q$ (where $q = p^n$ for a prime $p$), the algebraic closure $\overline{\mathbb{F}_q}$ is the union of all finite fields of characteristic $p$. It is an infinite field, and its absolute Galois group is isomorphic to the profinite completion of the integers, $\hat{\mathbb{Z}}$.
• p-adic Numbers: The algebraic closure of the field of $p$-adic numbers $\mathbb{Q}_p$ is denoted as $\overline{\mathbb{Q}_p}$. Unlike $\mathbb{C}$ over $\mathbb{R}$, $\overline{\mathbb{Q}_p}$ is not metrically complete. Its metric completion is denoted as $\mathbb{C}_p$, which is both algebraically closed and complete.

Separable Closure

Within the algebraic closure $\overline{F}$, there exists a unique maximal separable extension of $F$, known as the separable closure of $F$, denoted as $F^{sep}$. An algebraic extension is separable if the minimal polynomial of every element has distinct roots.

If the base field $F$ is a perfect field (which includes all fields of characteristic zero and all finite fields), then every algebraic extension is separable, and thus the separable closure coincides with the algebraic closure ($F^{sep} = \overline{F}$). For imperfect fields, such as the field of rational functions over a finite field $\mathbb{F}_p(t)$, the separable closure is a proper subfield of the algebraic closure. The separable closure is particularly important in Galois theory, as the absolute Galois group is formally defined as the automorphism group of the separable closure over the base field.

Cardinality and Automorphisms

The cardinality of the algebraic closure $\overline{F}$ is strictly determined by the cardinality of the base field $F$. If $F$ is a finite field or a countably infinite field (like $\mathbb{Q}$), its algebraic closure is countably infinite. If $F$ is uncountable (like $\mathbb{R}$ or $\mathbb{C}(x)$), the cardinality of $\overline{F}$ is exactly the same as the cardinality of $F$.

The group of all field automorphisms of $\overline{F}$ that fix every element of $F$ is called the absolute Galois group of $F$, denoted as $\text{Gal}(\overline{F}/F)$ or $G_F$. This group is a profinite group and encodes deep arithmetic and geometric information about the field $F$. For example, the absolute Galois group of $\mathbb{R}$ is the cyclic group of order 2, while the absolute Galois group of $\mathbb{Q}$ is a highly complex and deeply studied object in modern number theory.

Applications

The concept of algebraic closure is ubiquitous in advanced mathematics. In algebraic geometry, working over an algebraically closed field ensures that geometric objects (varieties) have points, which is essential for foundational results like Hilbert's Nullstellensatz. In Galois theory, the algebraic (or separable) closure provides the universal domain in which all polynomial equations can be solved, allowing the translation of field-theoretic problems into group-theoretic ones. In model theory, the theory of algebraically closed fields of a fixed characteristic is a prime example of a complete, uncountably categorical, and strongly minimal theory, serving as a cornerstone for geometric model theory.