In field theory, a separable extension is an algebraic extension L/K of fields where every element \alpha \in L is separable over K, meaning the minimal polynomial of \alpha over K has distinct roots in a splitting field.[1] This contrasts with inseparable extensions, which occur when some minimal polynomials have multiple roots.[2]The foundation of separability lies in the notion of separable polynomials: a nonzero polynomial f(X) \in K[X] is separable if it has no multiple roots in its splitting field, equivalently, if \gcd(f, f') = 1 in K[X], where f' is the formal derivative.[1] An algebraic element \alpha over K is separable if its minimal polynomial is separable, and since irreducible polynomials are separable precisely when their derivative is nonzero, all irreducible polynomials over fields of characteristic zero are separable.[3]Separable extensions are ubiquitous in characteristic zero, where every algebraic extension is separable, as the derivative condition holds trivially for all polynomials.[2] In characteristic p > 0, separability depends on the field: a field K is perfect if every algebraic extension is separable, which holds if K = K^p (every element is a p-th power); finite fields and their extensions are always separable, but purely inseparable extensions like K(\alpha) where \alpha^p \in K but \alpha \notin K exist otherwise.[2] For any extension L/K, one can decompose the degree [L:K] into separable degree [L:K]_s (the number of K-embeddings of L into an algebraic closure) and inseparable degree [L:K]_i = [L:K]/[L:K]_s, a power of p.[1]Finite separable extensions have particularly nice properties: they admit a primitive element, meaning L = K(\gamma) for some \gamma \in L, and the number of distinct K-embeddings equals the degree [L:K].[1] More generally, an algebraic extension is separable if it is generated by separable elements, and the compositum of separable extensions remains separable.[3] These concepts are central to Galois theory, where separable extensions ensure the Galois group acts faithfully on roots without fixed multiplicities.[1]
Fundamental Notions
Informal Discussion
In field theory, a separable extension arises as a fundamental concept distinguishing certain algebraic extensions of fields based on the nature of their minimal polynomials. Consider a field extension E/F, where F is the base field and E contains algebraic elements over F. An element \alpha \in E is called separable over F if its minimal polynomial over F has distinct roots in a splitting field, meaning the polynomial and its formal derivative are coprime. This condition ensures that the extension behaves "nicely" without multiple roots complicating the structure, a property that holds automatically for all algebraic extensions in characteristic zero fields like \mathbb{Q} or \mathbb{R}.[4][5]The notion of separability extends to the entire field E/F being separable if every \alpha \in E is separable over F. In characteristic p > 0, inseparability can occur when polynomials like X^p - a (with a not a p-th power in F) have multiple roots, leading to extensions where the degree exceeds the number of automorphisms. For instance, over the field \mathbb{F}_p(t) of rational functions in characteristic p, adjoining \sqrt{t} yields an inseparable extension of degree p, as the minimal polynomial X^p - t has a single root with multiplicity p. Such extensions lack the full symmetry expected in Galois theory, highlighting separability's role in ensuring the extension admits a primitive element and aligns with permutation actions on roots.[4][5]Informally, separable extensions capture the intuitive idea of fields generated by "simple" adjunctions of roots without degeneracy, facilitating tools like the primitive element theorem, which states that every finite separable extension is simple (generated by a single element). This theorem underscores separability's utility: for example, \mathbb{Q}(\sqrt{2})/\mathbb{Q} is separable and simple, with two automorphisms corresponding to the distinct roots of X^2 - 2. In contrast, inseparable cases force more complex constructions, often requiring transcendental elements to exhibit inseparability. Perfect fields—those where every algebraic extension is separable, such as finite fields—provide a clean setting where all irreducibles are separable, emphasizing separability's centrality in algebraic number theory and Galois representations.[4][6]
Separable and Inseparable Polynomials
In field theory, a polynomial f(x) \in K over a field K is defined to be separable if it has no multiple roots in an algebraic closure of K, meaning all its roots are simple. Equivalently, f(x) is separable if it is coprime to its formal derivative f'(x) in K, i.e., \gcd(f(x), f'(x)) = 1. This criterion holds because a multiple root \alpha of f(x) satisfies both f(\alpha) = 0 and f'(\alpha) = 0, implying a common factor. In fields of characteristic zero, a polynomial is separable if and only if it is square-free, meaning it has no repeated irreducible factors. This is equivalent to gcd(f, f') = 1, and while f' is nonzero for non-constant f, the coprimality detects the absence of multiple roots.[3][1][2]A polynomial is inseparable if it has at least one multiple root. Inseparable polynomials arise primarily in positive characteristic p > 0, where the derivative may vanish; specifically, an irreducible polynomial f(x) is inseparable if and only if f'(x) = 0, which occurs when f(x) is a polynomial in x^p, i.e., f(x) = g(x^p) for some g(y) \in K. More generally, any inseparable irreducible polynomial can be expressed as f(x) = g(x^{p^k}) where g(y) is separable and k \geq 1; here, the separable degree is \deg g and the inseparable degree is p^k. For example, over \mathbb{F}_p(t), the polynomial x^p - t is irreducible and inseparable, with a single root of multiplicity p in its splitting field. In contrast, x^2 - 2 over \mathbb{Q} is separable, as its roots \pm \sqrt{2} are distinct.[3][1][2]The separability of polynomials directly informs the separability of algebraic elements in field extensions. An algebraic element \alpha over K is separable if its minimal polynomial over K is separable; otherwise, it is inseparable. Inseparable polynomials give rise to inseparable elements and purely inseparable extensions, in which every element \alpha has a minimal polynomial over K of separable degree 1, i.e., with only one distinct root in its splitting field. Every finite extension decomposes uniquely into a separable part followed by a purely inseparable part, with degrees multiplying to the total degree.[3][1][2][7]
Definitions and Properties
Separable Elements and Separable Extensions
A separable polynomial over a field K is a nonzero polynomial f(X) \in K[X] that has distinct roots in a splitting field, meaning all roots have multiplicity one.[1] Equivalently, f(X) is separableif and only if it is coprime to its formal derivative f'(X).[1] In characteristic zero, every irreducible polynomial is separable, as the derivative is nonzero for nonconstant polynomials.[8] In positive characteristic p, an irreducible polynomial is inseparable if it is of the form g(X^p) for some g \in K[X], since then f'(X) = 0.[1]An algebraic element \alpha over K is separable if its minimal polynomial over K is separable.[1] For instance, \sqrt{2} is separable over \mathbb{Q} because its minimal polynomial X^2 - 2 has distinct roots \pm \sqrt{2}.[9] In contrast, over \mathbb{F}_p(t^p), the element t is inseparable, with minimal polynomial X^p - t^p = (X - t)^p, which has a multiple root.[8]A finite field extension L/K is separable if every element of L is separable over K.[1] This is equivalent to the extension degree [L:K] equaling the number of distinct K-algebra homomorphisms from L to an algebraic closure of K.[9] Finite extensions in characteristic zero are always separable.[8] Separability is preserved under towers: if E/L and L/K are separable, then E/K is separable.[1] Moreover, a finite separable extension is simple, meaning L = K(\alpha) for some separable \alpha \in L.[9] For example, \mathbb{Q}(\sqrt{2}, \sqrt{3})/\mathbb{Q} is separable of degree 4 and equals \mathbb{Q}(\sqrt{2} + \sqrt{3}).[9]The concept of separable extensions was introduced by Ernst Steinitz in 1910 as "extensions of the first kind," with the term "separable" later adopted by Bartel van der Waerden in his 1930 textbook Moderne Algebra.[1]
Separable Extensions within Algebraic Extensions
In the context of algebraic field extensions, a field extension L/K is termed separable if it is algebraic and every element \alpha \in L is separable over K, meaning the minimal polynomial of \alpha over K has distinct roots in a splitting field.[10][11] Equivalently, L/K is separable if the number of distinct K-embeddings of L into an algebraic closure \overline{K} equals the degree [L:K].[12] This contrasts with purely inseparable extensions, where the minimal polynomial of each generator has multiple roots, typically arising in positive characteristic p > 0. For instance, in characteristic p, the extension K(t)/(t^p - u) over K(u) is purely inseparable if u is transcendental.[10][11]A fundamental property of separable extensions is their transitivity: if L/K is separable and M/L is separable, then M/K is separable.[12][13] Moreover, every subextension of a separable extension is separable, ensuring that separability is preserved under intermediate fields.[13] The separable degree [L:K]_s, defined as the degree of the maximal separable subextension, is multiplicative over towers: [L:K]_s = [L:M]_s \cdot [M:K]_s.[12] In fields of characteristic zero, all algebraic extensions are separable, as the derivative criterion for multiple roots fails to produce inseparable polynomials. Similarly, perfect fields—those where every algebraic extension is separable, such as finite fields or characteristic zero fields—exhibit no inseparable behavior.[11][12]Finite separable extensions admit a primitive element theorem: there exists \theta \in L such that L = K(\theta), and the extension is simple.[12][11] For normal separable extensions, the Galois group \mathrm{Gal}(L/K) acts transitively on the roots, establishing a bijection with the intermediate fields via the fundamental theorem of Galois theory.[13] A key criterion for separability in positive characteristic is linear disjointness from the p-th power extension K^{1/p}/K: L/K is separable if and only if L \otimes_K K^{1/p} \cong L^{1/p} \times \cdots \times L^{1/p} (with [K:K^p] factors).[12] This framework underpins further developments in algebraic number theory, where separable extensions ensure the ring of integers remains finitely generated.[10]
Specialized Contexts and Criteria
Separability of Transcendental Extensions
In field theory, the notion of separability, originally defined for algebraic extensions, extends to general field extensions K/k that may include transcendental elements. A key concept is that of a separably generated extension: K/k is separably generated if there exists a transcendence basis \{x_i \mid i \in I\} for K/k such that K is a separable algebraic extension of k(x_i \mid i \in I).[14] More broadly, K/k is called separable if every finitely generated subextension K'/k is separably generated. This ensures that the transcendental and algebraic components interact in a way that preserves separability properties locally.[14]Purely transcendental extensions provide a fundamental example. If K = k(x_1, \dots, x_n) where each x_i is transcendental over k and algebraically independent, then \{x_1, \dots, x_n\} serves as a transcendence basis, and K/k(x_1, \dots, x_n) is the trivial algebraic extension of degree 1, which is separable. Thus, purely transcendental extensions are always separable, regardless of the characteristic of k.[15] In characteristic zero, every field extension is separable, as the tensor product k \otimes_k K is reduced (i.e., has no nonzero nilpotents), a condition equivalent to separability for arbitrary extensions.[15] In positive characteristic, however, transcendental extensions may fail separability if the algebraic closure introduces inseparability, but purely transcendental ones remain separable.[15]A related refinement is the separating transcendence basis, particularly useful for finitely generated extensions over perfect fields. For a finitely generated extension K/k with k perfect (e.g., algebraically closed or of characteristic zero), there exists a transcendence basis \{y_1, \dots, y_m\} such that K/k(y_1, \dots, y_m) is a finite separable algebraic extension.[16] This basis "separates" the transcendental structure from potential inseparability in the algebraic part, allowing decomposition of the extension into separable components. For instance, in characteristic p > 0, if K = k(x) with x transcendental, then \{x\} is separating, as the extension over it is separable. Such bases exist by induction on the number of generators, leveraging the perfectness of k to avoid inseparable factors.[16]These definitions highlight that transcendental extensions are separable when their algebraic hull over a transcendence basis avoids multiple roots in minimal polynomials. Subextensions of separable extensions are also separable, ensuring closure under composition.[14] In applications, such as geometric algebra, separability of transcendental extensions corresponds to geometrically reduced schemes, linking field-theoretic properties to algebraic geometry.[14]
Differential Criteria
In the context of field extensions, differential criteria for separability leverage the module of Kähler differentials, which provides an algebraic analogue of differential forms adapted to commutative rings. For a field extension K/F, the Kähler differentials \Omega_{K/F} form a K-module equipped with a universal derivation d: K \to \Omega_{K/F} that is F-linear and satisfies the Leibniz rule d(ab) = a\, db + b\, da for a, b \in K, with d(f) = 0 for f \in F. This module is generated by symbols d\alpha for \alpha \in K, subject to relations arising from the ring structure.[17][18]A key differential criterion states that if K/F is a finite algebraic extension of fields, then K/F is separable if and only if \Omega_{K/F} = 0 as a K-module. To see the forward direction, suppose K = F(\alpha) for a primitive element \alpha with separable minimal polynomial f(x) \in F, so f'(\alpha) \neq 0. The relation d(f(\alpha)) = 0 yields f'(\alpha) \, d\alpha = 0, implying d\alpha = 0 since K is a field and f'(\alpha) \neq 0. As \Omega_{K/F} is generated by d\alpha, it vanishes. For general finite separable extensions, the primitive element theorem reduces to this case. Conversely, if \Omega_{K/F} = 0, the separability follows from the fact that non-separable elements would generate non-zero differentials, as in the case of purely inseparable extensions where \Omega_{K/F} \neq 0.[17][18]This criterion extends to finitely generated algebraic extensions: K/F is separable if and only if \Omega_{K/F} = 0. For instance, in characteristic p > 0, a purely inseparable extension like F(\alpha) where \alpha^p \in F but \alpha \notin F yields \Omega_{K/F} \cong K \, d\alpha \neq 0, confirming inseparability. In contrast, transcendental extensions have \Omega_{K/F} free of rank equal to the transcendence degree, highlighting the distinction from algebraic separability. These results underpin broader applications in algebraic geometry, where vanishing differentials characterize smooth or étale morphisms.[18]