1 Basic definitions
An algebraic extension is a field extension whose elements all satisfy polynomial equations over the smaller field. This idea provides a way to measure how closely a larger field is tied to its base field through polynomial relations. It is a central concept in field theory because it separates extensions built from roots of polynomials from those involving genuinely new, transcendental behavior.
1.1 Field extensions
A field extension consists of two fields, with one contained in the other, so that the operations of addition, subtraction, multiplication, and division in the smaller field agree with those in the larger one. The smaller field is called the base field, and the larger field is often viewed as obtained by adding new elements to it. Many constructions in algebra begin with such extensions, especially when one wants to study polynomial roots inside a larger ambient field.
1.2 Algebraic elements
An element of a field extension is called algebraic over the base field if it is a root of some nonzero polynomial with coefficients in that base field. If no such polynomial exists, the element is transcendental. Algebraic elements are the building blocks of algebraic extensions, since an extension is algebraic precisely when all of its elements are algebraic over the base field.
1.2.1 Minimal polynomials
For an algebraic element, there is a unique monic polynomial of least degree in the base field having that element as a root. This polynomial is called the minimal polynomial. It is irreducible over the base field, and it captures the essential algebraic dependence of the element on the base field.
1.2.2 Degree of an algebraic element
The degree of an algebraic element is the degree of its minimal polynomial. This number measures the complexity of adjoining the element to the base field. In many cases, it also equals the dimension of the field obtained by adjoining the element, viewed as a vector space over the base field.
1.3 Algebraic extensions
A field extension is algebraic if every element of the larger field is algebraic over the smaller field. Such extensions arise naturally when one adjoins roots of polynomials and then closes under field operations. They are fundamental in the study of splitting fields, finite extensions, and algebraic closures.
1.3.1 Equivalent characterizations
An extension is algebraic if and only if each element lies in a finite-dimensional simple extension of the base field. Another useful characterization is that the extension is a union of finite subextensions generated by algebraic elements. These equivalent viewpoints are often used to shift between elementwise and structural arguments.
1.3.2 Examples and non-examples
The field obtained by adjoining the square root of 2 to the rational numbers is algebraic, since the new element satisfies x^2 - 2 = 0. More generally, number fields are algebraic extensions of the rational numbers. By contrast, the field of rational functions in one variable over a field is typically not algebraic over that field, because the variable itself does not satisfy any nonzero polynomial relation with coefficients in the base field.
2 Fundamental properties
Algebraic extensions behave well under many standard field constructions. They are closed under taking subextensions and combining algebraic fields inside a common overfield. Their structure is especially manageable when the extension is finitely generated, since such extensions are often finite.
2.1 Closure properties
Basic operations on algebraic extensions usually preserve algebraicity. If one starts with algebraic elements and forms fields from them, the resulting field remains algebraic over the original base. This stability makes the theory highly usable in iterative constructions.
2.1.1 Subfields of algebraic extensions
Any intermediate field between a base field and an algebraic extension is itself algebraic over the base field. Indeed, every element of the intermediate field already lies in the larger algebraic extension, so it satisfies a polynomial over the base field. This fact is frequently used to analyze chains of fields one step at a time.
2.1.2 Composites of algebraic extensions
When two algebraic extensions sit inside a common field, the field they generate together is again algebraic over the original base. The composite extension contains sums, products, and inverses formed from algebraic elements, and these remain algebraic. This closure property supports constructions involving multiple roots or several adjoining steps.
2.2 Finitely generated algebraic extensions
If an extension is generated by finitely many algebraic elements, then it is finite over the base field. This is a major structural result, since it converts an elementwise algebraic condition into a finite-dimensional vector space statement. Such extensions are among the most important objects in classical field theory.
2.2.1 Finite extensions
A finite extension is one whose dimension as a vector space over the base field is finite. Every finite extension is algebraic, because each element satisfies a nontrivial linear dependence relation over the base field, which can be converted into a polynomial relation. Finite extensions appear throughout the theory of splitting fields and algebraic number fields.
2.2.2 Towers of extensions
When fields are arranged in a chain, the degree of the whole extension can often be computed by multiplying the degrees of the successive steps. This is known as the tower law. It is especially useful for breaking complicated extensions into simpler pieces generated by one element at a time.
2.3 Degrees of extensions
The degree of a field extension is its dimension as a vector space over the base field. For algebraic extensions, this degree may be finite or infinite, but it remains a fundamental invariant. It reflects how large the extension is and often controls the behavior of adjoining elements.
2.3.1 Multiplicativity of degree
In a tower of finite extensions, the degree of the top field over the bottom field equals the product of the degrees of the intermediate steps. This multiplicative rule is one of the most useful computational tools in field theory. It allows one to infer the size of an extension from simpler ones built in stages.
2.3.2 Basis considerations
A finite algebraic extension has a basis over the base field, and every element can be written uniquely as a linear combination of basis vectors. For a simple extension generated by one algebraic element, powers of that element often form a convenient spanning set. Basis descriptions make explicit calculations possible in arithmetic and polynomial theory.
3 Construction and examples
Algebraic extensions are commonly built by adjoining roots of polynomials to a field. This process produces a wide range of examples, from quadratic fields to more intricate splitting fields. Classical constructions also connect algebraic extensions with geometric questions and the study of special numbers.
3.1 Simple algebraic extensions
A simple algebraic extension is generated by a single algebraic element. Such extensions are among the most accessible examples, since their structure is determined by one polynomial relation. Many important fields in algebra and number theory arise in this way.
3.1.1 Adjoining a root of a polynomial
If a polynomial over a field has a root in some larger field, one can enlarge the base field by adjoining that root. The resulting extension contains the original field and the chosen root, together with all expressions obtained from them using field operations. This method is the standard starting point for constructing algebraic fields.
3.1.2 Quotient field descriptions
A simple extension generated by a root of an irreducible polynomial can often be described as a quotient of a polynomial ring by the ideal generated by that polynomial. In this model, the class of the indeterminate represents the chosen root. This representation gives an explicit algebraic description of the extension and is widely used in computations.
3.2 Splitting fields
A splitting field of a polynomial is the smallest field extension in which the polynomial factors completely into linear terms. Such fields are always algebraic over the base field, since they are generated by the polynomial’s roots. They play a central role in the transition from polynomial equations to Galois theory.
3.2.1 Roots of polynomials
The roots of a polynomial may lie outside the original field, and adjoining all of them produces an extension where the polynomial becomes fully reducible. The resulting field contains every algebraic relation needed to express the polynomial completely. This viewpoint makes splitting fields a natural setting for studying equation solving.
3.2.2 Algebraic closures
An algebraic closure of a field is an algebraic extension in which every nonconstant polynomial has a root, and hence every polynomial splits completely. Such a closure is large enough to contain all algebraic elements over the base field. It provides a universal environment for working with polynomial equations.
3.3 Classical examples
Several familiar number systems and special extensions illustrate the theory of algebraic extensions in concrete form. These examples show how algebraic elements arise in arithmetic, geometry, and the study of roots of unity.
3.3.1 Quadratic extensions
Quadratic extensions are generated by a root of a degree-two irreducible polynomial. They are among the simplest nontrivial algebraic extensions and often have a basis of the form 1, α. Their structure is easy to compute and serves as a model for more general finite extensions.
3.3.2 Cyclotomic extensions
Cyclotomic extensions are obtained by adjoining roots of unity to a field, especially the rational numbers. They are algebraic because roots of unity satisfy equations of the form x^n - 1 = 0. These extensions are important in the study of periodicity, integer arithmetic, and classical polynomial factorization.
3.3.3 Constructible numbers
Constructible numbers arise from geometric constructions using a straightedge and compass. They form an algebraic extension of the rational numbers built through successive quadratic steps. Their study connects algebraic extensions with classical problems of Euclidean geometry.
4 Relations to other classes of extensions
Algebraic extensions are best understood by contrasting them with transcendental extensions and by examining their interaction with normality and separability. These relations help organize field extensions into a broader framework. They also prepare the ground for the deeper structure of Galois theory.
4.1 Transcendental extensions
An extension is transcendental when it contains at least one element not algebraic over the base field. Such extensions are structurally different from algebraic ones and typically resemble fields of rational functions. The distinction is fundamental in separating polynomially constrained elements from free generators.
4.1.1 Algebraic versus transcendental elements
An algebraic element satisfies some polynomial equation over the base field, while a transcendental element satisfies none. The two types of elements are mutually exclusive and often determine the nature of an extension. Many fields can be analyzed by decomposing them into algebraic and transcendental parts.
4.1.2 Purely transcendental extensions
A purely transcendental extension is generated by one or more algebraically independent transcendental elements. It behaves in some respects like a field of rational functions in several variables. Such extensions contrast sharply with algebraic ones, since no new element is forced to obey a polynomial relation over the base field.
4.2 Normal and separable extensions
Normality and separability are additional conditions imposed on algebraic extensions. They refine the classification by describing how polynomials split and how roots relate to one another. Together, they are essential in the formulation of Galois theory.
4.2.1 Normality in algebraic extensions
An algebraic extension is normal when every irreducible polynomial over the base field that has one root in the extension splits completely there. This condition ensures that the extension contains all conjugates of its algebraic elements. Normal extensions are the natural habitat of splitting fields.
4.2.2 Separable extensions
An algebraic element is separable if its minimal polynomial has distinct roots in a splitting field. An extension is separable if all of its elements are separable over the base field. Separability avoids repeated-root pathologies and is especially important when the field’s characteristic influences polynomial behavior.
4.3 Simple extension theorem
Under suitable conditions, finitely generated algebraic extensions can be generated by a single element. This result, often called the primitive element theorem, significantly simplifies the study of many finite fields. It shows that complicated collections of algebraic adjunctions may be compressed into one generator.
4.3.1 Primitive element theorem
The primitive element theorem states that a finite separable extension can be written as a simple extension. In other words, there exists a single element whose adjoining produces the whole field. This theorem is a powerful simplification tool in algebraic field theory.
4.3.2 Conditions for simplicity
Not every finite extension is automatically simple, but many are simple under mild hypotheses. Separability is the key condition in the most common form of the theorem. When simplicity holds, computations and theoretical arguments become much more transparent.
5 Advanced topics
Beyond the basic theory, algebraic extensions lead naturally to algebraic closures, automorphism groups, and infinite towers of fields. These topics connect field theory to deeper structural results and to the modern formulation of Galois correspondence. They also show how algebraic extensions organize large classes of fields in a unified way.
5.1 Algebraic closure
An algebraic closure is a maximal algebraic extension in which every polynomial has a root. It can be viewed as a complete setting for polynomial factorization and root-adjoining. Such closures are indispensable in abstract algebra because they provide a universal algebraic environment.
5.1.1 Existence
Every field has an algebraic closure. The proof typically uses maximality arguments from set theory or constructs a sufficiently large field containing roots of all polynomials. This existence result ensures that any field can be embedded into an algebraically complete setting.
5.1.2 Uniqueness up to isomorphism
Although an algebraic closure is not unique as a literal field, any two algebraic closures of the same base field are isomorphic over that field. This means that the choice of closure does not affect the essential algebraic structure. As a result, one may speak of “the” algebraic closure up to canonical equivalence.
5.2 Galois theory connections
Algebraic extensions provide the framework in which Galois theory relates field automorphisms to polynomial equations. The interaction between field structure and symmetry becomes especially clear for normal and separable extensions. This correspondence is one of the major achievements of abstract algebra.
5.2.1 Fixed fields and automorphism groups
For a field extension, one can study the group of automorphisms that preserve the base field. The elements fixed by all such automorphisms form a subfield, called the fixed field. This relationship captures the symmetry of an algebraic extension in a form suitable for classification.
5.2.2 Fundamental theorem of Galois theory
For finite Galois extensions, intermediate fields correspond to subgroups of the Galois group in an inclusion-reversing way. This theorem reveals a deep link between field theory and group theory. It allows many questions about field extensions to be translated into questions about symmetry groups.
5.3 Infinite algebraic extensions
Some algebraic extensions are not finite-dimensional, even though every element remains algebraic over the base field. These infinite extensions arise as unions of increasingly large finite extensions. They are common in areas where one adjoins many algebraic elements at once.
5.3.1 Direct limits
An infinite algebraic extension can often be described as a direct limit of finite subextensions. In this approach, the larger field is assembled from a directed family of smaller fields. This viewpoint is useful for handling extensions built from infinitely many stages.
5.3.2 Algebraic extensions of infinite degree
An algebraic extension may have infinite degree as a vector space over the base field. Even then, each individual element still satisfies a polynomial relation. Such extensions retain the algebraic character of their elements while being too large to be finite-dimensional.