Simple extension

In field theory, a simple extension of a field F is a field extension K/F generated by adjoining a single element \alpha \in K, denoted K = F(\alpha), which is the smallest field containing both F and \alpha.^[1] This construction forms the foundation for studying algebraic structures, where elements of K can be expressed as rational functions in \alpha with coefficients in F. Simple extensions are classified into two primary types based on the nature of the adjoined element \alpha. If \alpha is algebraic over F, meaning it satisfies a non-constant polynomial equation with coefficients in F, then F(\alpha) is a finite-dimensional vector space over F with dimension equal to the degree of the minimal polynomial of \alpha, and \{1, \alpha, \dots, \alpha^{n-1}\} serves as a basis, where n is that degree.^[1] In this case, F(\alpha) is isomorphic to the quotient ring F/(m(x)), where m(x) is the monic irreducible minimal polynomial of \alpha.^[2] Conversely, if \alpha is transcendental over F, it satisfies no such polynomial, and F(\alpha) is an infinite-dimensional extension isomorphic to the field of rational functions F(x) in an indeterminate x.^[1] A key result concerning simple extensions is the primitive element theorem, which states that every finite separable extension of fields admits a simple extension representation, i.e., there exists a primitive element \alpha such that the extension is F(\alpha).^[3] This theorem is fundamental in Galois theory, enabling the analysis of symmetries and solvability of polynomials through the structure of simple extensions. Simple extensions also play a crucial role in constructing splitting fields and studying algebraic closures, as they allow the adjunction of roots of irreducible polynomials to build larger fields.^[3]

Fundamentals

Definition

In field theory, a field extension L/K is termed simple if there exists an element \theta \in L such that L = K(\theta).^[2] Here, K(\theta) denotes the smallest field containing both the base field K and the element \theta, obtained by adjoining \theta to K.^[2] The field K(\theta) is explicitly defined as the quotient field (or field of fractions) of the polynomial ring K[\theta], which consists of all finite linear combinations \sum c_i \theta^i with coefficients c_i \in K.^[4] As such, the elements of K(\theta) are all rational expressions in \theta with coefficients from K, taking the form

\frac{\sum_{i=0}^n a_i \theta^i}{\sum_{j=0}^m b_j \theta^j},

where a_i, b_j \in K, the sums are finite, and the denominator is nonzero.^[4] This construction assumes familiarity with the general notion of field extensions, where K is a subfield of L.^[1]

Primitive Elements

In field theory, a primitive element for a simple extension L/K is an element \theta \in L such that L = K(\theta), meaning the entire extension is generated by adjoining \theta to the base field K.^[5] By the definition of a simple extension, at least one primitive element exists, but not every element of L qualifies as primitive; only those \theta for which the subfield generated by K and \theta coincides with L do so.^[6] Primitive elements are not unique for a given simple extension. If \theta is primitive, then \theta + c for any c \in K is also primitive, though the structure of the extension relative to different primitives may vary, such as differing minimal polynomials in algebraic settings. The simple extension K(\theta) is constructed as the smallest field containing K and \theta, comprising all elements of the form p(\theta)/q(\theta), where p(x), q(x) \in K are polynomials and q(\theta) \neq 0. This set is closed under field operations: addition and multiplication follow from those on polynomials and quotients, while every non-zero element has an inverse given by reciprocal quotients.^[1]

Classification

Algebraic Simple Extensions

An algebraic simple extension is a field extension L/K where L = K(\theta) for some element \theta \in L that is algebraic over the base field K. This means \theta satisfies a non-zero polynomial equation p(X) \in K[X], i.e., p(\theta) = 0 with p not the zero polynomial.^[2] The minimal polynomial of \theta over K, denoted \mu_\theta(X), is the unique monic irreducible polynomial in K[X] of least degree that has \theta as a root. This polynomial plays a central role in characterizing the extension, as L consists precisely of the elements of the form \sum_{i=0}^{d-1} a_i \theta^i where d = \deg(\mu_\theta) and a_i \in K, with the relation \mu_\theta(\theta) = 0 allowing reduction of higher powers.^[1] The degree of the extension [L : K] equals the degree of the minimal polynomial, [L : K] = \deg(\mu_\theta) = n < \infty, which implies that L is a finite-dimensional vector space over K of dimension n. This finite degree distinguishes algebraic simple extensions from their transcendental counterparts and ensures that every element of L is algebraic over K.^[2] Not all algebraic simple extensions are separable. In characteristic p > 0, inseparability arises if the minimal polynomial \mu_\theta(X) has multiple roots in an algebraic closure of K, which occurs precisely when \mu_\theta(X) and its formal derivative \mu_\theta'(X) share a common root (or equivalently, when \mu_\theta'(X) = 0, so \mu_\theta(X) = Q(X^p) for some Q \in K[X]). For example, adjoining a p-th root of an element without one in K yields such an inseparable extension.^[7]

Transcendental Simple Extensions

A transcendental simple extension of a field K is a field extension L = K(\theta) where \theta is transcendental over K, meaning that no non-zero polynomial with coefficients in K vanishes at \theta.^[8] In this case, the evaluation map \varphi: K[X] \to K[\theta] given by \sum a_i X^i \mapsto \sum a_i \theta^i is an isomorphism of rings, with trivial kernel, confirming the transcendental nature of \theta.^[8] Consequently, the degree of the extension [L : K] is infinite, as the powers \{1, \theta, \theta^2, \dots \} form a linearly independent set over K.^[9] Such extensions are isomorphic to the field of rational functions K(X) over K in one indeterminate X.^[10] Specifically, there exists a field isomorphism \phi: K(X) \to K(\theta) that fixes K pointwise and sends X to \theta, mapping rational functions p(X)/q(X) to p(\theta)/q(\theta), where p, q \in K[X] and q \neq 0.^[8] This isomorphism arises because both fields are the fraction fields of isomorphic polynomial rings K[X] and K[\theta], preserving the structure of rational expressions.^[10] In a transcendental simple extension, \theta behaves as a free indeterminate over K, imposing no algebraic relations from K.^[9] Elements of L are thus rational functions in \theta with coefficients in K, and the extension lacks the polynomial constraints characteristic of algebraic extensions, allowing for unbounded independence in the field structure.^[8]

Structure and Properties

Basis Representation

In the algebraic case, consider a simple extension L = K(\theta) where \theta is algebraic over the base field K with minimal polynomial \mu_\theta(X) \in K[X] of degree n = [L : K]. The set \{1, \theta, \theta^2, \dots, \theta^{n-1}\} forms a K-basis for L as a vector space over K, known as the power basis.^[11] Every element \alpha \in L can be uniquely expressed as a K-linear combination \alpha = \sum_{i=0}^{n-1} a_i \theta^i with a_i \in K.^[11] This basis property arises from the division algorithm in the polynomial ring K[X]: for any f(X) \in K[X], there exist unique q(X), r(X) \in K[X] such that f(X) = q(X) \mu_\theta(X) + r(X) with \deg r < n, so f(\theta) = r(\theta) and \deg r < n.^[11] Linear independence follows because if \sum_{i=0}^{n-1} a_i \theta^i = 0 with not all a_i = 0, then \mu_\theta(X) would divide the nonzero polynomial \sum_{i=0}^{n-1} a_i X^i of degree less than n, a contradiction.^[11] Consequently, \dim_K L = n = \deg \mu_\theta.^[11] In contrast, for a simple transcendental extension L = K(\theta) where \theta is transcendental over K, the extension has infinite degree [L : K] = \infty, so no finite K-basis exists.^[12] Here, L is isomorphic to the field of rational functions K(X), and its elements are formal quotients p(\theta)/q(\theta) with p, q \in K[X], q \neq 0, without restriction to finite support in a basis expansion.^[11]

Isomorphism Theorems

In the context of simple field extensions, the isomorphism theorems provide a fundamental characterization of the structure of L = K(\theta) as a quotient of the polynomial ring K[X] or its fraction field, depending on whether \theta is algebraic or transcendental over K. These results rely on the evaluation homomorphism and the first isomorphism theorem for rings.^[13]^[14] For the algebraic case, suppose \theta is algebraic over K with minimal polynomial \mu_\theta(X) \in K[X], which is the monic irreducible polynomial of least degree having \theta as a root (as discussed in the section on algebraic simple extensions). The evaluation homomorphism \phi: K[X] \to L is defined by \phi(f) = f(\theta) for all f \in K[X]. This map is a ring homomorphism, and its kernel is the principal ideal (\mu_\theta(X)), since \mu_\theta is the minimal polynomial. By the first isomorphism theorem for rings, L \cong K[X] / (\mu_\theta(X)), where the isomorphism sends the coset X + (\mu_\theta(X)) to \theta. This quotient is a field because (\mu_\theta(X)) is maximal, as \mu_\theta is irreducible.^[13]^[14] In the transcendental case, \theta has no minimal polynomial over K, so the evaluation homomorphism \phi: K[X] \to L given by \phi(f) = f(\theta) has trivial kernel, making it injective. Here, K[X] embeds into L, and since L is the smallest field containing K and \theta, L is the fraction field of K[X] under this embedding. Thus, L \cong K(X), the field of rational functions in one indeterminate over K, with the isomorphism sending X to \theta. The kernel remains trivial, confirming that no non-constant polynomial in K[X] vanishes at \theta.^[13]^[14]

Key Theorems and Examples

Primitive Element Theorem

A finite field extension L/K is simple if and only if there are only finitely many intermediate fields between K and L. More specifically, the primitive element theorem states that every finite separable extension of fields is simple, meaning there exists a primitive element \theta \in L such that L = K(\theta).^[6] In fields of characteristic zero, all finite extensions are separable, so the theorem implies that every finite extension in characteristic zero is simple.^[6] In positive characteristic p > 0, separability of a finite extension requires that the minimal polynomials of its generators (beyond the first) are separable, meaning they have distinct roots in an algebraic closure or splitting field.^[6] A proof sketch proceeds by induction on the degree of the extension. For a separable extension L = K(\alpha_1, \dots, \alpha_n) with [\alpha_i : K] < \infty and \alpha_i separable over K for i \geq 2, assume the result holds for fewer generators and consider L = F(\alpha, \beta) where F = K(\alpha_1, \dots, \alpha_{n-1}). If the base field F is infinite, elements of the form \theta = \alpha + c \beta with c \in F generate L except for finitely many "bad" values of c, which would otherwise create intermediate fields containing proper subextensions; separability ensures the minimal polynomial of \beta over F(\theta) has full degree for suitable c. The full proof leverages the finiteness of the automorphism group of the normal closure to bound intermediate fields.^[6] A key corollary is that every finite normal separable extension is simple, as normality implies the extension is Galois, whose finite Galois group yields only finitely many subgroups and thus finitely many intermediate fields by the Fundamental Theorem of Galois Theory.^[15]

Illustrative Examples

Simple algebraic extensions provide foundational illustrations of how adjoining a single root of an irreducible polynomial over a base field yields a finite-degree extension. For instance, the field of complex numbers \mathbb{C} is a simple extension of the real numbers \mathbb{R} generated by i, where the minimal polynomial of i over \mathbb{R} is \mu_i(X) = X^2 + 1. This extension has degree 2, with basis \{1, i\} over \mathbb{R}.^[16] Similarly, the extension \mathbb{Q}(\sqrt{2})/\mathbb{Q} is simple, generated by \sqrt{2} with minimal polynomial \mu(X) = X^2 - 2, also of degree 2.^[17] Extensions generated by multiple elements can often be expressed as simple extensions using a primitive element, as in the case of \mathbb{Q}(\sqrt{2}, \sqrt{3})/\mathbb{Q}, which has degree 4. This extension is simple, generated by the primitive element \alpha = \sqrt{2} + \sqrt{3}, whose minimal polynomial over \mathbb{Q} is (X^2 - 5)^2 - 24 = X^4 - 10X^2 + 1, obtained via resultants or elimination methods.^[17]^[18] In contrast, transcendental simple extensions are infinite-degree and arise when adjoining an element with no minimal polynomial over the base field. A canonical example is the field of rational functions \mathbb{R}(X)/\mathbb{R}, generated by the indeterminate X, which is transcendental over \mathbb{R}. Elements here are quotients of polynomials in X with real coefficients, forming a field of transcendence degree 1.^[19] Finite fields also admit simple algebraic extensions. The extension \mathbb{F}_{p^n}/\mathbb{F}_p of degree n is simple, generated by a root \alpha of any irreducible polynomial of degree n over \mathbb{F}_p. For example, \mathbb{F}_4/\mathbb{F}_2 is generated by a root of the irreducible polynomial X^2 + X + 1 over \mathbb{F}_2, yielding four elements: $0, 1, \alpha, \alpha + 1.^[16]^[20]