a branch of algebra, an algebraic field extension E / F {\displaystyle E/F} is called a separable extension if for every α ∈ E {\displaystyle \alpha...

closure of K. Since a separable extension of a separable extension is again separable, there are no finite separable extensions of Ksep, of degree > 1...

of a separable field extension Separable differential equation, in which separation of variables is achieved by various means Separable extension, in field...

In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed...

extension is a field extension that is both normal and separable. A consequence of the primitive element theorem states that every finite separable extension...

Separable polynomials are used to define separable extensions: A field extension K ⊂ L is a separable extension if and only if for every α in L which is algebraic...

a separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field extension. A homomorphism...

that p doesn't divide n, since otherwise this can fail even to be a separable extension). In general, however, the Galois groups of n-th roots of elements...

transcendence basis S such that L is a separable algebraic extension over K(S). A field extension L / K is said to be separably generated if it admits a separating...

coefficients in k. Integral element Lüroth's theorem Galois extension Separable extension Normal extension Fraleigh (2014), Definition 31.1, p. 283. Malik, Mordeson...

of α {\displaystyle \alpha } over F is not a separable polynomial. If F is any field, the trivial extension F ⊇ F {\displaystyle F\supseteq F} is purely...

condition for a separable extension of a Hilbertian field to be Hilbertian. Let K be a Hilbertian field and L a separable extension of K. Assume there...

integrally closed domain with field of fractions K. If L/K is a finite separable extension, then the integral closure A ′ {\displaystyle A'} of A in L is a...

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

field K, making a torus over S an algebraic group whose extension to some finite separable extension L is a finite product of copies of Gm/L. In general,...

theory, the primitive element theorem states that every finite separable field extension is simple, i.e. generated by a single element. This theorem implies...

extensions F / E, which are, by definition, those that are separable and normal. The primitive element theorem shows that finite separable extensions...

of categories between the finite unramified extensions of a local field K and finite separable extensions of the residue field of K. Again, let L / K...

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...

any field extension F/k. Every irreducible polynomial over k has non-zero formal derivative. Every irreducible polynomial over k is separable. Every finite...

is closely related to separability. A unital associative algebra A over a field k is said to be separable if the base extension A ⊗ k F {\displaystyle...

the United States was increased by 20 years under the Copyright Term Extension Act. This legislation was the subject of substantial criticism following...

Galois extensions E / F {\displaystyle E/F} for a fixed field. The inverse limit is denoted Gal ⁡ ( F ¯ / F ) := lim ← E / F  finite separable ⁡ Gal ⁡...

{\displaystyle x^{p^{n}}-x=0} .[citation needed] Any finite field extension of a finite field is separable and simple. That is, if E {\displaystyle E} is a finite...

field EH is a normal extension of F (or, equivalently, Galois extension, since any subextension of a separable extension is separable) if and only if H is...

{\displaystyle R} is an integral domain and L {\displaystyle L} a finite separable extension of K {\displaystyle K} , then the integral closure S {\displaystyle...

is a finite extension field of k. The variety X is smooth over k if and only if E is a separable extension of k. Thus, if E is not separable over k, then...

it is separable). A splitting field of a set P of polynomials is the smallest field over which each of the polynomials in P splits. An extension L that...

Field extension Algebraic extension Splitting field Algebraically closed field Algebraic element Algebraic closure Separable extension Separable polynomial...

with field of fractions K and integral closure of B in a finite separable extension L of K. (this implies that B is also a Dedekind domain.) Let I A...