In mathematics, a polynomial P(X) over a given field K is separable if its roots are distinct in an algebraic closure of K, that is, the number of distinct...

a separable extension if for every α ∈ E {\displaystyle \alpha \in E} , the minimal polynomial of α {\displaystyle \alpha } over F is a separable polynomial...

Separable permutation, a permutation that can be obtained by direct sums and skew sums of the trivial permutation Separable polynomial, a polynomial whose...

every polynomial is a product of separable polynomials (since every polynomial is a product of its irreducible factors, and these are separable over a...

polynomial time whether a given separable permutation is a pattern in a larger permutation, or to find the longest common subpattern of two separable...

E/F} is a normal extension and a separable extension. E {\displaystyle E} is a splitting field of a separable polynomial with coefficients in F . {\displaystyle...

separable closure of K. Since a separable extension of a separable extension is again separable, there are no finite separable extensions of Ksep, of degree...

In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence ( x n ) n = 1 ∞ {\displaystyle...

additive polynomials are an important topic in classical algebraic number theory. Let k be a field of prime characteristic p. A polynomial P(x) with...

factor which is not separable (i.e., the irreducible factor is a polynomial in x p {\displaystyle x^{p}} ). The discriminant of a polynomial is, up to a scaling...

In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: signal processing as Hermitian wavelets...

\alpha \in E\setminus F} , the minimal polynomial of α {\displaystyle \alpha } over F is not a separable polynomial. If F is any field, the trivial extension...

extension generated by roots of separable polynomials. Perfect field A field such that every finite extension is separable. All fields of characteristic...

Every irreducible polynomial over k is separable. Every finite extension of k is separable. Every algebraic extension of k is separable. Either k has characteristic...

In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after optical physicist Frits Zernike...

over the finite field with p elements (see Separable polynomial: the point here is that P is not separable). If L is the field extension K(T 1/p) (the...

irreducible polynomial p ( x ) = ( x − a ) ∑ i = 0 n − 1 b i x i {\textstyle p(x)=(x-a)\sum _{i=0}^{n-1}b_{i}x^{i}} , then a separability idempotent is...

code belongs to the class of maximum distance separable codes. While the number of different polynomials of degree less than k and the number of different...

{\displaystyle K} ; that is, an element that is not a root of any univariate polynomial with coefficients in K {\displaystyle K} . In other words, a transcendental...

because if α and β are roots of the cubic polynomial, we shall have (α/β)3 =1 and the cubic is a separable polynomial. Then L/K is a Kummer extension. More...

Hilbert space is separable provided it contains a dense countable subset. Along with Zorn's lemma, this means a Hilbert space is separable if and only if...

group in degree five. It is a polynomial of degree 6. These three resolvents have the property of being always separable, which means that, if they have...

introduced the subject for studying roots of polynomials. This allowed him to characterize the polynomial equations that are solvable by radicals in terms...

non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since...

L / K {\displaystyle L/K} is called separable if the minimal polynomial of every element of L over K is separable, i.e., has no repeated roots in an algebraic...

extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so...

Algebraically closed field Algebraic element Algebraic closure Separable extension Separable polynomial Normal extension Galois extension Abelian extension Transcendence...

the polynomial equation x p n − x = 0 {\displaystyle x^{p^{n}}-x=0} .[citation needed] Any finite field extension of a finite field is separable and simple...

even on an uncountable supporting set, giving an example of non-separable polynomially reflexive Banach space. Distortion problem Sequence space, Schauder...

E[X] / f(X), where f is an irreducible polynomial (as above). For such an extension, being normal and separable means that all zeros of f are contained...