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...

{\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...

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 ⁡...

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...

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...

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

because there exist Azumaya algebras, which are trivial over a finite separable extension, but not over the base field. Special groups were defined in 1958...

closed field is regular. An extension is regular if and only if it is separable and primary. A purely transcendental extension of a field is regular. There...

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...