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Splitting field

In field theory, a splitting field of a f(x) over a base field K is the smallest L of K in which f(x) factors completely into a product of linear factors, and L is generated by K and the roots of f(x). This extension is minimal in the sense that no proper subfield of L containing K allows f(x) to split fully. Such splitting fields always exist for any nonconstant over a , constructed by successively adjoining roots of irreducible factors until the polynomial splits. They are unique up to : any two splitting fields of the same polynomial over K are isomorphic as extensions of K, with the isomorphism fixing K pointwise, and the degree of the extension equals the number of such isomorphisms when the polynomial is separable. For example, the splitting field of x^2 + 1 over \mathbb{R} is \mathbb{C}, while that of x^4 - 2 over \mathbb{Q} is \mathbb{Q}(\sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=4&&&citation_type=wikipedia}}{2}, i), which has degree 8. Splitting fields play a central role in , where the Galois group of the extension—comprising the automorphisms of the splitting field fixing the base field—captures the symmetries of the roots and determines solvability by radicals for polynomials over fields of characteristic zero. In cases of separable polynomials, the order of this group equals the degree of the extension, linking algebraic structure to . They also arise naturally in the study of finite fields and algebraic closures, providing the foundation for understanding field extensions and irreducibility.

Fundamentals

Definition

In field theory, a splitting field of a non-constant f \in K over a K is the smallest L/K such that f factors completely into linear factors in L, that is, f(X) = c \prod_{i=1}^n (X - \alpha_i) for some constant c \in K and roots \alpha_i \in L, with n = \deg(f). This concept generalizes to a of polynomials \{f_1, \dots, f_m\} \subset K: the splitting field is the smallest extension L/K in which each f_j splits completely into linear factors over L. Here, K is assumed to be a and each f (or f_j) non-constant; moreover, L is generated over K by of f (or all the f_j). Such an L is algebraic over K, and if f has finite degree, then L is finite-dimensional over K; specifically, for a separable of degree n, the degree [L : K] divides n!. Splitting fields play a central role in by providing a minimal extension in which to study the roots of polynomials, avoiding larger unnecessary field extensions.

Basic Properties

A splitting field L of a polynomial f \in K over a K is unique up to as a K-; specifically, any two splitting fields L and L' of f over K are K- via an isomorphism that maps the roots of f in L to the corresponding roots in L'. This isomorphism preserves the field structure and fixes K pointwise, ensuring that the algebraic relations among the roots are maintained. The extension L/K is normal, meaning that every irreducible polynomial in K that has at least one root in L splits completely into linear factors in L. In fact, a finite extension is normal if and only if it is a splitting field of some polynomial over the base field. Regarding separability, if f is separable—meaning it has distinct roots in its splitting field—then L/K is a separable extension. Moreover, in characteristic zero, every splitting field extension is separable, as all algebraic extensions in this case lack inseparable elements. If \deg(f) = n < \infty, then L/K is a finite extension with [L : K] \leq n!. The roots \alpha_1, \dots, \alpha_m of f (counting multiplicities) each satisfy their respective minimal polynomials over K, and L is generated as the smallest field containing K and all these roots, so L = K(\alpha_1, \dots, \alpha_m). Additionally, the Galois group \mathrm{Gal}(L/K), when L/K is Galois, acts transitively on the roots of each irreducible factor of f over K.

Construction

Motivation

The concept of the splitting field emerged in the early 19th century as part of Évariste Galois's groundbreaking work on the solvability of polynomial equations by radicals, motivated by the need to determine when roots could be expressed using finite sequences of arithmetic operations and root extractions. Galois, in his 1831 memoir submitted to the Paris Academy, introduced the idea of field extensions that adjoin all roots of a polynomial, providing a framework to analyze the symmetries among those roots through associated permutation groups. This approach resolved longstanding questions, such as the impossibility of solving general quintic equations by radicals, by linking the structure of these extensions to group-theoretic properties. Practically, splitting fields address the need to study the roots of polynomials in a minimal algebraic setting, avoiding the construction of larger structures like algebraic closures, which contain roots of all polynomials over the base field but are often infinite and unwieldy for specific computations. By focusing on the smallest extension where a given polynomial factors completely into linear factors, splitting fields facilitate factorization and root-finding while preserving the base field's characteristics, such as being of finite degree over fields like the rationals. This minimalism enables precise analysis of polynomial behavior without unnecessary elements, making it essential for applications in algebraic number theory and beyond. In broader algebraic theory, splitting fields serve as foundational components for Galois groups, which encode the automorphisms of the extension and classify its subextensions, thereby revealing the symmetries inherent in polynomial roots. The Galois group of a splitting field over the base field acts transitively on the roots of each irreducible factor, providing a permutation representation that captures the extension's structure and solvability conditions. This connection underpins much of modern Galois theory, where splitting fields of separable polynomials yield normal extensions whose groups determine key properties like separability and normality. A significant application arises in the inverse Galois problem, which asks whether every finite group can be realized as the Galois group of the splitting field of some polynomial over the rationals, highlighting the role of these fields in embedding arbitrary group structures into field extensions. Unlike the algebraic closure, which is infinite and its full automorphism group is not computable in a finite sense, splitting fields are finite extensions tailored to individual polynomials, allowing explicit computation of their Galois groups and enabling progress on realizing specific groups, such as symmetric or alternating groups. This specificity makes splitting fields indispensable for targeted investigations in group theory and field extensions.

Iterative Adjoining of Roots

The standard method for constructing the splitting field of a non-constant polynomial f(x) \in K over a field K proceeds by iteratively adjoining roots of its irreducible factors, yielding a tower of simple algebraic extensions. Begin with the base field K_0 = K. Factor f(x) = c \prod_{i=1}^m g_i(x)^{e_i}, where c \in K^\times, the g_i(x) are distinct monic irreducible polynomials in K, and the e_i are positive integers representing multiplicities. Select one irreducible factor, say g_1(x), and adjoin a root \alpha_1 of g_1 to form the simple extension K_1 = K_0(\alpha_1) \cong K_0/(g_1(x)), which has degree [K_1 : K_0] = \deg g_1. Over K_1, the polynomial f(x) now has \alpha_1 as a root, so it factors further; repeat the process by adjoining a root \alpha_2 of a remaining irreducible factor of the updated f(x) to obtain K_2 = K_1(\alpha_2), and continue iteratively through K_3 = K_2(\alpha_3), ..., K_r = K_{r-1}(\alpha_r), until the final field L = K_r in which f(x) factors completely into linear terms. For a separable polynomial f, the factorization has e_i = 1 for all i (distinct irreducible factors with no multiple roots), and the iterative adjoining preserves separability throughout the tower: each simple extension K_j / K_{j-1} is separable since \alpha_j is a simple root of its minimal polynomial, and the composite extension remains separable. In this case, at each step, f(x) acquires exactly one new linear factor (x - \alpha_j) over K_j, simplifying the remaining factorization. If f is inseparable (possible in positive characteristic), multiplicities e_i > 1 may persist, but the process still adjoins roots of the irreducible factors, though the extension may involve purely inseparable components; however, the splitting is defined via complete splitting regardless. The of the full extension satisfies the tower law: [L : K] = \prod_{j=1}^r [K_j : K_{j-1}], where each [K_j : K_{j-1}] is the of the minimal of \alpha_j over K_{j-1} (the irreducible factor selected at that step). For separable f, this equals the order of the and divides (\deg f)!, reflecting the finite nature of the process for polynomials of finite . To establish minimality, note that L is generated over K by the full set of roots \{\alpha_1, \dots, \alpha_n\} of f (where n = \deg f), as the iterative process explicitly adjoins them all, and f splits completely in L. Any proper subfield of L containing K would fail to contain at least one root (by the simplicity and irreducibility at each adjoining step), hence would not split f, confirming L as the smallest such extension. The existence of this construction relies on the fact that every non-constant over K has at least one in some of K; iteratively, this allows adjoining until splitting occurs. More fundamentally, the existence of follows from the existence of an of K, proved via applied to the of all of K (ordered by inclusion), where every chain has an upper bound (the union), yielding a maximal that is algebraically closed and thus contains splitting fields for all polynomials. In characteristic zero, explicit constructions (e.g., via complex numbers) also suffice, but the general proof uses .

Quotient Ring Approach

The quotient ring approach to constructing splitting fields leverages the structure of polynomial rings over a base field K to adjoin roots algebraically, providing a foundation for proving existence and uniqueness without relying solely on transcendental methods. For a separable irreducible polynomial f(x) \in K of degree n \geq 2, form the quotient ring M = K / (f(x)). Since f is irreducible, the ideal (f(x)) is maximal, making M a field extension of K of degree n, isomorphic to the simple extension K(\alpha) where \alpha denotes the image of x in M, a root of f. To determine if M is the splitting field, examine the factorization of f in M. If f factors completely into distinct linear factors over M, then all roots lie in M, and M is the splitting field of f over K, as it is the smallest such extension containing K and all roots. Otherwise, f factors in M as a product of irreducibles of strictly lower degree (by properties of field extensions), and the construction proceeds by selecting an irreducible factor g(y) and forming the further quotient M / (g(y)), which adjoins an additional root and yields a larger field extension. This process terminates after finitely many steps, as degrees decrease, resulting in a splitting field. The quotient construction yields a field at each step since the polynomials selected are irreducible over the current field. In the inseparable case, the resulting extension is inseparable, but the splitting field is still obtained iteratively. In the general case of a separable but possibly reducible f(x) \in K, the splitting field L over K is the compositum of the splitting fields of the distinct irreducible factors of f. Abstractly, L is isomorphic to K(\alpha_1, \dots, \alpha_m), where the \alpha_i are the distinct of f in some \overline{K} of K; this field embeds into \overline{K}, and by the uniqueness of splitting fields up to isomorphism, the construction yields L as the final extension in the tower. For a direct non-iterative view, L can be realized as a of a multivariate K[x_1, \dots, x_m] by the of the evaluation map sending each x_i to a \alpha_i, though this is typically computed implicitly via the iterative . This method relates closely to simple extensions: each quotient step produces a simple algebraic extension isomorphic to adjoining a single root, building the full splitting field as a tower K \subset K(\alpha_1) \subset K(\alpha_1, \alpha_2) \subset \cdots \subset L, where each layer is obtained via a quotient ring. The quotient ring approach advantages lie in its ring-theoretic elegance, enabling existence proofs through algebraic geometry tools such as Krull's theorem, which guarantees maximal ideals containing principal ideals generated by irreducibles, thus ensuring field quotients. It is particularly valuable in for analyzing integral extensions and prime factorization via Dedekind's criterion, which uses quotient rings to study how primes ramify in splitting fields of number fields. Computationally, it is less direct for explicit splitting fields, as factoring over intermediate extensions requires additional effort compared to sequential root adjunction.

Examples

Complex Numbers over the Reals

The polynomial f(X) = X^2 + 1 in \mathbb{R}[X] serves as a fundamental example of a splitting field, illustrating how an irreducible over the reals leads to the complex numbers. This has no in \mathbb{R}, as the equation x^2 + 1 = 0 implies x = \pm \sqrt{-1}, which are not real; thus, f(X) is irreducible over \mathbb{R}. To construct the splitting field, adjoin a root i of f(X) to \mathbb{R}, yielding the extension \mathbb{C} = \mathbb{R}(i). This field is isomorphic to the quotient ring \mathbb{R}[X] / (X^2 + 1), where the coset X + (X^2 + 1) corresponds to i. In \mathbb{C}[X], the polynomial factors completely as f(X) = (X - i)(X + i), confirming that \mathbb{C} is the splitting field of f(X) over \mathbb{R}. The degree of this extension is [\mathbb{C} : \mathbb{R}] = 2, matching the degree of the f(X). The extension \mathbb{C}/\mathbb{R} is both , as all field extensions of characteristic zero are separable, and , since it is the splitting field of a separable polynomial. The \mathrm{Gal}(\mathbb{C}/\mathbb{R}) is isomorphic to \mathbb{Z}/2\mathbb{Z}, consisting of the identity automorphism and complex conjugation, which sends i to -i while fixing \mathbb{R}. This group acts transitively on \{i, -i\}. While \mathbb{C} is the algebraic closure of \mathbb{R}, meaning it splits every non-constant polynomial in \mathbb{R}[X], in this specific case it is the minimal such field for f(X).

Cubic Polynomials over the Rationals

A classic example of a splitting field for a cubic polynomial over the rationals is provided by the irreducible polynomial f(X) = X^3 - 2 \in \mathbb{Q}[X]. This polynomial is irreducible over \mathbb{Q} by the Eisenstein criterion with prime p=2. The roots of f(X) are \alpha = \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{2} (the real cube root of 2), \beta = \alpha \omega, and \gamma = \alpha \omega^2, where \omega = e^{2\pi i / 3} is a primitive cube root of unity. To construct the splitting field L of f(X) over \mathbb{Q}, first adjoin the real root \alpha, yielding the extension \mathbb{Q}(\alpha)/\mathbb{Q} of degree 3, since the minimal polynomial of \alpha over \mathbb{Q} is f(X) itself. Next, adjoin \omega to \mathbb{Q}(\alpha); the minimal polynomial of \omega over \mathbb{Q}(\alpha) is X^2 + X + 1, which is irreducible because \mathbb{Q}(\alpha) is a real field and thus does not contain the complex non-real \omega. Therefore, [\mathbb{Q}(\alpha, \omega) : \mathbb{Q}(\alpha)] = 2, and the total degree [L : \mathbb{Q}] = [L : \mathbb{Q}(\alpha)] \cdot [\mathbb{Q}(\alpha) : \mathbb{Q}] = 2 \cdot 3 = 6, where L = \mathbb{Q}(\alpha, \omega). This construction follows the iterative adjoining of roots, building the splitting field as a tower of simple extensions. The \mathrm{Gal}(L/\mathbb{Q}) is isomorphic to the S_3, which has order 6 matching the degree of the extension; it is non-abelian and acts transitively on the three \alpha, \beta, \gamma. This structure arises because the resolvent for the cubic has non-square in \mathbb{Q}, leading to the full S_3 rather than the alternating . In contrast, for irreducible cubics over \mathbb{Q} with three real roots, the splitting field may have smaller degree. For instance, consider g(X) = X^3 - 3X + 1, which is irreducible over \mathbb{Q} by the (possible rational roots \pm 1 do not work). Its is 81, a square in \mathbb{Q}, so the is the A_3 \cong \mathbb{Z}/3\mathbb{Z}, and the splitting field has degree 3 over \mathbb{Q} (adjoining one root generates the full field containing all three real roots). For reducible cubics, the splitting field degree is at most 2, as a linear factor provides a rational root, and the quadratic factor splits over a quadratic extension if irreducible. These cases illustrate how the possibilities for cubics over \mathbb{Q} are limited to subgroups of S_3, with the splitting field degree dividing 6.

Cyclotomic Polynomials

The nth cyclotomic polynomial \Phi_n(X) is defined as the monic polynomial whose roots are the primitive nth roots of unity, that is, \Phi_n(X) = \prod (X - \zeta), where the product runs over all primitive nth roots of unity \zeta. These polynomials have integer coefficients and are irreducible over the field of rational numbers \mathbb{Q}, a result originally proved by Gauss. The splitting field of \Phi_n(X) over \mathbb{Q} is the \mathbb{Q}(\zeta_n), where \zeta_n is a nth . This extension has degree [\mathbb{Q}(\zeta_n) : \mathbb{Q}] = \varphi(n), where \varphi(n) denotes , which counts the number of integers from 1 to n that are coprime to n. The irreducibility of \Phi_n(X) ensures that \mathbb{Q}(\zeta_n) is the minimal such extension, and adjoining any nth generates all roots of \Phi_n(X). A concrete example occurs for n=5, where \Phi_5(X) = X^4 + X^3 + X^2 + X + 1. The splitting field is L = \mathbb{Q}(\zeta_5), with [L : \mathbb{Q}] = 4 since \varphi(5) = 4. The \mathrm{Gal}(L/\mathbb{Q}) is isomorphic to (\mathbb{Z}/5\mathbb{Z})^\times \cong \mathbb{Z}/4\mathbb{Z}. Cyclotomic extensions \mathbb{Q}(\zeta_n)/\mathbb{Q} are Galois extensions with abelian Galois groups, specifically \mathrm{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q}) \cong (\mathbb{Z}/n\mathbb{Z})^\times. This abelian structure makes them fundamental examples in , where every finite abelian extension of \mathbb{Q} is contained in some cyclotomic extension. The X^n - 1 factors over \mathbb{Q} as X^n - 1 = \prod_{d \mid n} \Phi_d(X), where the product is over all positive divisors d of n. Consequently, the splitting field of X^n - 1 over \mathbb{Q} is precisely \mathbb{Q}(\zeta_n), as it contains all nth of unity and is generated by any primitive one.

Finite Field Extensions

In finite fields, which have positive characteristic p, splitting fields play a central role in constructing field extensions. Let \mathbb{F}_q denote the finite field with q = p^k elements for some prime p and integer k \geq 1. Every finite extension of \mathbb{F}_q is a splitting field of some irreducible polynomial over \mathbb{F}_q, and in particular, every finite field \mathbb{F}_{q^m} is the splitting field of the polynomial X^{q^m} - X over \mathbb{F}_q. This polynomial is separable and factors completely into distinct linear factors in \mathbb{F}_{q^m}, with exactly q^m roots corresponding to all elements of the field. A concrete example illustrates this construction. Consider the polynomial f(X) = X^2 + X + 2 over \mathbb{F}_3, the field with three elements \{0, 1, 2\}. This polynomial is irreducible over \mathbb{F}_3 because it has no roots in \mathbb{F}_3: substituting the elements yields f(0) = 2, f(1) = 1 + 1 + 2 = 1 \neq 0, and f(2) = 4 + 2 + 2 = 1 + 2 + 2 = 2 \neq 0 (modulo 3). Adjoining a root \alpha of f(X) gives the extension \mathbb{F}_9 = \mathbb{F}_3(\alpha), a field with nine elements satisfying \alpha^2 + \alpha + 2 = 0. The other root is \alpha^3 = 2\alpha + 2 (since the Frobenius map \beta \mapsto \beta^3 sends roots to roots in characteristic 3), so f(X) splits completely as (X - \alpha)(X - (2\alpha + 2)) in \mathbb{F}_9[X]. The degree of the extension is [\mathbb{F}_9 : \mathbb{F}_3] = 2. More generally, for an irreducible polynomial f(X) of degree n over \mathbb{F}_q, its splitting field is the unique (up to isomorphism) finite field \mathbb{F}_{q^n}, obtained by adjoining one root and containing all others via powers of the Frobenius automorphism. The Galois group \mathrm{Gal}(\mathbb{F}_{q^n}/\mathbb{F}_q) is cyclic of order n and generated by the Frobenius map \phi_q: x \mapsto x^q. In characteristic p, a f(X) is separable its formal f'(X) is not identically zero, ensuring no multiple roots in the splitting field. Since finite fields are perfect (every is separable), all irreducible polynomials over \mathbb{F}_q are separable. Splitting fields over finite fields find applications in and . For instance, Reed-Solomon codes, widely used for error correction in storage and communication systems, are constructed using evaluations of polynomials over extension fields \mathbb{F}_{q^n}, which serve as splitting fields of irreducible polynomials of degree n over \mathbb{F}_q.