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Separable polynomial

In algebra, a separable polynomial over a field K is a nonzero polynomial f(X) \in K[X] that has distinct roots in a splitting field over K, meaning it has no repeated roots.^[1] Equivalently, f is separable if the greatest common divisor of f and its formal derivative f' is a unit in K[X], i.e., \gcd(f, f') = 1.^[2] This condition ensures that the polynomial factors into distinct linear factors in an algebraic closure of K.^[3] Separable polynomials play a central role in Galois theory, where the splitting field of a separable polynomial over K is a Galois extension, allowing the fundamental theorem of Galois theory to apply fully.^[4] In fields of characteristic zero, every irreducible polynomial is separable, as the derivative of a nonzero polynomial is never zero.^[1] However, in characteristic p > 0, inseparability can occur when the derivative vanishes, such as for polynomials of the form X^p - a where a \in K is not a p-th power; this phenomenon is exclusive to positive characteristic and leads to purely inseparable extensions.^[1] An algebraic element \alpha over K is called separable if its minimal polynomial is separable, and a field extension is separable if every element in it is separable over the base field.^[5] The concept of separability extends to broader notions in algebraic geometry and number theory, such as étale algebras and separable extensions of rings, but its foundational importance lies in distinguishing "nice" behavior in field extensions where Galois groups act freely without fixed points from repeated roots.

Definitions

Modern definition

In modern algebra, a polynomial f(x) \in K over a field K is defined to be separable if, in a splitting field of f over K, it factors as a product of distinct linear factors, meaning all roots are simple (i.e., have multiplicity one) and there are no multiple roots.^[1] This definition emphasizes the distinctness of roots in an algebraic closure of K. An equivalent characterization is that f(x) is separable if and only if its discriminant \Delta(f), defined as the product \prod_{i < j} (\alpha_i - \alpha_j)^2 over the roots \alpha_1, \dots, \alpha_n in a splitting field (up to a sign and leading coefficient factor), is nonzero as an element of K.^[6] For monic polynomials, this discriminant lies directly in K, and its nonvanishing confirms the absence of multiple roots without needing to adjoin the roots explicitly. This formulation is particularly useful for computational verification. A concrete example is the polynomial x^2 - 2 \in \mathbb{Q}, which is separable because its roots \sqrt{2} and -\sqrt{2} in the splitting field \mathbb{Q}(\sqrt{2}) are distinct.^[1] More generally, for any separable polynomial f \in K, the splitting field L of f over K is a Galois extension of K, meaning L/K is both normal and separable, with the Galois group acting transitively on the roots.^[7]

Older definition

In older treatments, such as those referenced by Nathan Jacobson, a polynomial was considered separable if each of its irreducible factors over K has distinct roots in a splitting field (i.e., no multiple roots within each irreducible factor).^[8] This differed from the modern definition, which requires all roots of the polynomial to be distinct globally, even if the polynomial is reducible. For example, under the older definition, (x-1)^2 would be separable since its irreducible factor x-1 has a simple root, but it is inseparable under the modern definition due to the multiple root. This older perspective aligned with the modern one for irreducible polynomials and fields of characteristic zero but highlighted differences for reducible cases. The concept evolved in the late 19th and early 20th centuries through works like Richard Dedekind's supplements to Dirichlet's Vorlesungen über Zahlentheorie (1877), which advanced Galois theory for number fields, and later abstract treatments that unified separability across characteristics using criteria like the gcd with the derivative.^[1]

Properties and criteria

Criterion via derivatives

A polynomial f(x) \in K over a field K, where \deg f \geq 1, is separable if and only if \gcd(f(x), f'(x)) = 1 in K, with f'(x) denoting the formal derivative of f(x).^[9]^[10]^[1] To see why this criterion holds, consider the splitting field L of f(x) over K. Write f(x) = \prod_{i=1}^n (x - \alpha_i)^{m_i} in L, where the \alpha_i are the roots (counted without multiplicity) and each m_i \geq 1. The polynomial f(x) is separable precisely when all m_i = 1, meaning it has distinct roots in L. A multiple root occurs if and only if some m_j > 1, in which case x - \alpha_j divides both f(x) and f'(x) (since the derivative satisfies f'( \alpha_j ) = 0 when m_j > 1). Thus, f(x) and f'(x) share a common root in L if and only if f(x) has a multiple root. Applying the Euclidean algorithm in K, which remains valid upon extension to L since L is a field extension, it follows that \gcd(f(x), f'(x)) has positive degree in K if and only if f(x) and f'(x) share a common irreducible factor, hence a common root in L. Therefore, \gcd(f(x), f'(x)) = 1 if and only if f(x) is separable.^[10]^[1] This connection to the factorization explicitly links the criterion to the definition: separability requires all exponents m_i = 1 in the splitting, which is equivalent to no shared factors with the derivative.^[9] For example, consider f(x) = x^2 + 1 \in \mathbb{Q}. The formal derivative is f'(x) = 2x, and \gcd(x^2 + 1, 2x) = 1 since x^2 + 1 is irreducible over \mathbb{Q} and has no linear factors. Thus, f(x) is separable, with distinct roots \pm i in \mathbb{C}. In characteristic p > 0, take f(x) = x^2 + x + 1 \in \mathbb{F}_2. Here, f'(x) = 1 (as the derivative of x^2 vanishes but that of x is 1), so \gcd(f(x), 1) = 1, confirming separability; the roots lie in \mathbb{F}_4 and are distinct.^[10] In fields of characteristic p > 0, the criterion remains valid, but f'(x) = 0 can occur for nonconstant f(x) if all exponents in f(x) are multiples of p, implying f(x) is a p-th power in K and hence inseparable (as \gcd(f(x), 0) = f(x) \neq 1). No additional tests are required beyond computing the gcd.^[9]^[1]

Behavior in characteristic zero

In fields of characteristic zero, every irreducible polynomial over the field is separable. This key theorem arises from the behavior of the formal derivative: for any non-constant polynomial f, the derivative f' is a non-zero polynomial of strictly lower degree, and since the field has no prime characteristic, f and f' share no common roots in a splitting field, ensuring \gcd(f, f') = 1. Thus, f has no multiple roots, making it separable by definition.^[1]^[11] As a direct consequence, every irreducible polynomial factors into distinct linear factors in a splitting field. General polynomials factor uniquely into products of (possibly repeated) irreducible polynomials, each of which is separable. This contrasts with positive characteristic, where irreducible polynomials can be inseparable.^[1] This property extends to field extensions: all algebraic extensions of a field of characteristic zero, such as extensions of the rational numbers \mathbb{Q} or the real numbers \mathbb{R}, are separable. In particular, every element in such an extension has a separable minimal polynomial over the base field. For instance, the cyclotomic polynomials \Phi_n(x) over \mathbb{Q}, which are irreducible and generate cyclotomic extensions, are separable due to the characteristic zero setting.^[1]^[11] The universal separability in characteristic zero played a crucial role in the early development of Galois theory, as Évariste Galois formulated his ideas over fields like \mathbb{Q}, where inseparability issues do not arise, allowing focus on permutation groups and solvability by radicals without additional complications from multiple roots.^[12]

Behavior in positive characteristic

In fields of positive characteristic p > 0, the behavior of separable polynomials deviates significantly from the characteristic zero case, as irreducible polynomials can admit repeated roots in their splitting fields, leading to inseparability. A key issue arises when an irreducible polynomial is a p-th power in some sense; for instance, the polynomial x^p - t over the function field k(t), where k is a field of characteristic p, is irreducible but inseparable, having a single root of multiplicity p in its splitting field.^[1] To quantify this, the separable degree of a polynomial f over a field of characteristic p is defined as the degree of its largest separable subpolynomial, obtained by extracting the maximal p-power factors from its irreducible components; if f is separable, this equals the full degree of f. For an irreducible polynomial f(x), it can be uniquely written as f(x) = g(x^{p^e}) where g is separable and irreducible, and the separable degree is then \deg g.^[13] A concrete example illustrates potential inseparability: consider polynomials of the form x^p - a over a field extension of \mathbb{F}_p that is not perfect; for certain a not in the image of the Frobenius map, such polynomials can be irreducible and inseparable. More generally, purely inseparable polynomials, which generate purely inseparable extensions, are characterized by having zero derivative (f' = 0) and being perfect p-th powers in their splitting fields, such as f(x) = (x - \alpha)^ {p^m} for some \alpha and integer m \geq 1.^[1]

Relation to field extensions

Separable extensions

A field extension L/K is defined to be separable if every element \alpha \in L has a separable minimal polynomial over K, where a polynomial is separable if it has distinct roots in a splitting field over K.^[1] Equivalently, L/K is separable if L = K(S) for some set S consisting of elements that are separable over K.^[1] This definition aligns with the modern notion of separable polynomials, as an algebraic extension is separable precisely when it is generated by roots of separable irreducible polynomials.^[1] For finite extensions, L/K is separable if and only if the separable degree [L:K]_s, which counts the number of K-embeddings of L into an algebraic closure \bar{K} of K, equals the full degree [L:K].^[2] In this case, the primitive element theorem guarantees that L = K(\gamma) for some primitive element \gamma \in L whose minimal polynomial over K is separable.^[1] A concrete example is the extension obtained by adjoining a root of a separable irreducible polynomial to K; since the minimal polynomial has distinct roots, the resulting simple extension is separable.^[1] For instance, \mathbb{Q}(\sqrt{2}) is separable over \mathbb{Q} because the minimal polynomial x^2 - 2 has distinct roots.^[1] A key characterization of separability is that L/K is separable if and only if the tensor product L \otimes_K \bar{K} is a product of fields (equivalently, it is reduced, containing no nonzero nilpotent elements).^[14] This condition ensures that the extension has no inseparable components, as the tensor product decomposes into copies of \bar{K} corresponding to the embeddings without nilpotent radicals.^[14] In the finite normal case, where L/K is a Galois extension, the order of the Galois group |\mathrm{Gal}(L/K)| equals the degree [L:K].^[1]

Inseparable extensions

An inseparable field extension L/K is one that is not separable, meaning it contains at least one element whose minimal polynomial over K has a multiple root in an algebraic closure.^[15] Such extensions arise exclusively in fields of positive characteristic p > 0, where the Frobenius endomorphism plays a central role.^[1] A purely inseparable extension is a special case where every element \alpha \in L is purely inseparable over K, meaning the minimal polynomial of \alpha over K is of the form x^{p^m} - a for some a \in K and integer m \geq 0.^[16] In such extensions, the minimal polynomials have derivative zero, leading to multiple roots and precluding the existence of a non-trivial Galois group, as the extension cannot be normal.^[17] Any finite field extension L/K of characteristic p > 0 admits a unique decomposition as a tower K \subseteq M \subseteq L, where M/K is the maximal separable subextension (obtained as the separable closure of K in L) and L/M is purely inseparable of p-power degree.^[18] More precisely, if e is the minimal integer such that L \subseteq M(K^{p^e}) for the separable closure M of K in L, then L/M is purely inseparable of degree p^e.^[15] A classic example is the extension L = k(t^{1/p}) / K = k(t) over a field k of characteristic p > 0, where t is transcendental over k; here, L/K is purely inseparable of degree p, with minimal polynomial x^p - t over K, which factors as (x - t^{1/p})^p in L.^[19] Inseparable extensions are absent in perfect fields, which are either of characteristic zero or of characteristic p > 0 where every element has a p-th root in the field; algebraically closed fields are perfect, ensuring all their algebraic extensions are separable.^[20]^[17]

Applications

Role in Galois theory

In Galois theory, separable polynomials play a foundational role by ensuring that the associated field extensions are separable, which is a prerequisite for the full applicability of the fundamental theorem of Galois theory. Specifically, a Galois extension is defined as a field extension that is both normal and separable; the separability condition guarantees that the Galois group acts faithfully on the roots of the minimal polynomial, as the distinct roots allow for a transitive and faithful permutation representation without fixed points arising from multiplicities. This faithful action enables the bijection between subfields of the extension and subgroups of the Galois group, as well as the correspondence between intermediate fields and quotient groups.^[1]^[4] For an irreducible separable polynomial f \in K, the splitting field of f over K is a Galois extension, serving as the normal closure of the extension K(\alpha)/K where \alpha is any root of f. This follows because the splitting field is normal (the polynomial splits completely) and separable (due to the distinct roots of f), allowing the Galois group to be computed as the group of automorphisms fixing K and permuting the roots transitively. In characteristic zero, every algebraic extension is separable since all irreducible polynomials have distinct roots, making Galois theory applicable without restriction to inseparability issues.^[4]^[1]^[21] The role of separability extends to applications like solvability by radicals, where a separable polynomial f \in K is solvable by radicals if and only if its Galois group over K is solvable; this criterion relies on the separability of resolvent polynomials in the radical tower, ensuring the Galois groups remain manageable. The unsolvability of the general quintic equation over \mathbb{Q} illustrates this, as its Galois group is the symmetric group S_5, which is non-solvable, tying the impossibility directly to the structure of Galois groups arising from separable polynomials in characteristic zero. For example, the polynomial x^4 + 1 \in \mathbb{Q} is separable with distinct roots, and its splitting field over \mathbb{Q} has Galois group isomorphic to the Klein four-group \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}, which acts faithfully by permuting the four roots.^[22]^[23]^[24]

Finite fields and Artin-Schreier theory

Finite fields \mathbb{F}_{p^n} are Galois extensions of the prime field \mathbb{F}_p with cyclic Galois group generated by the Frobenius automorphism x \mapsto x^p. All irreducible polynomials over finite fields are separable, as finite fields are perfect: the Frobenius map is an automorphism, hence surjective, implying every element is a p-th power and thus every algebraic extension is separable.^[20] In fields of characteristic p, Artin-Schreier theory describes cyclic Galois extensions of degree p as those obtained by adjoining a root of an Artin-Schreier polynomial x^p - x - a = 0, where a lies outside the image of the map b \mapsto b^p - b. These polynomials are separable whenever irreducible, since the resulting extension is Galois of prime degree p.^[25] Primitive polynomials over \mathbb{F}_p, defined as the minimal polynomials of primitive elements (generators of the multiplicative group \mathbb{F}_{p^n}^\times), are irreducible and therefore separable. They are essential for explicit constructions of finite fields and underpin applications in coding theory, including the design of linear feedback shift registers for error-correcting codes, and in cryptography for efficient finite field computations in protocols like elliptic curve cryptography.^[26] A concrete example is the polynomial x^2 + x + 1 over \mathbb{F}_2, which is irreducible and separable, with roots generating the extension \mathbb{F}_4. Its derivative $1 \neq 0 confirms separability, and adjoining a root yields the field with elements \{0, 1, \omega, \omega+1\}, where \omega^2 = \omega + 1.^[7] In more advanced settings involving p-adic fields, Witt vectors construct integral lifts of the residue field that resolve inseparability issues by forming perfect rings, while separability remains crucial for analyzing Frobenius actions in the theory of local fields.^[27]

References

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Dec 6, 2017 · If we take a finite field F q with q = p n , it outputs a ring W p ( F q ) = O K where K is the unique unramified extension of Q p of degree n .