Separable extension - Citizendia

In mathematics, an algebraic field extension L/K is separable if it can be generated by adjoining to K a set each of whose elements is a root of a separable polynomial over K. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Abstract algebra, a Field extension L / K is called algebraic if every element of L is algebraic over K, i In Mathematics, a Polynomial P ( X) is separable over a field K if all of its irreducible factors have Distinct In that case, each β in L has a separable minimal polynomial over K. In Linear algebra, the minimal polynomial of an n -by- n matrix A over a field F is the Monic polynomial

The condition of separability is central in Galois theory. In Mathematics, more specifically in Abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory A perfect field is one for which all finite (equivalently, algebraic) extensions are separable. There exists a simple criterion for perfectness: a field F is perfect if and only if

F has characteristic 0, or
F has a nonzero characteristic p, and every element of F has a p-th root in F. In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's

Equivalently, the second condition says that the Frobenius endomorphism of F, $x\mapsto x^p$ , is an automorphism. In Commutative algebra and field theory, which are branches of Mathematics, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself

In particular, all fields of characteristic 0 and all finite fields are perfect. In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements This means that the separability condition can be assumed in many contexts. The effects of inseparability (necessarily for infinite K of characteristic p) can be seen in the primitive element theorem, and for the tensor product of fields. In Mathematics, more specifically in Field theory, the primitive element theorem provides a characterization of the Finite field extensions which are In Mathematics, the theory of fields in Abstract algebra lacks a Direct product: the direct product of two fields considered as a ring is

Given a finite extension L/K of fields, there is a largest subfield M of L containing K such that M is a separable extension of K. When M = K the extension L/K is called a purely inseparable extension. In general an algebraic extension factors as a purely inseparable extension of a separable extension, since the compositum of a family of separable extensions is again separable.

Purely inseparable extensions do occur for quite natural reasons, for example in algebraic geometry in characteristic p. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with If K is a field of characteristic p, and V an algebraic variety over K of dimension > 0, consider the function field K(V) and its subfield K(V)^p of p-th powers. This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety In Mathematics, more specifically in Abstract algebra, field extensions are the main object of study in field theory. This is always a purely inseparable extension. Such extensions occur as soon as one looks at multiplication by p on an elliptic curve over a finite field of characteristic p. In Mathematics, an elliptic curve is a smooth, projective Algebraic curve of genus one on which there is a specified point O

In dealing with non-perfect fields K, one introduces the separable closure K^sep inside an algebraic closure, which is the largest separable subextension of K^alg/K. In Mathematics, particularly Abstract algebra, an algebraic closure of a field K is an Algebraic extension of K that is Then Galois theory can be carried out inside K^sep.

References

Hungerford, Thomas (1974). Algebra. Springer. ISBN 0-387-90518-9.
Lang, Serge (2002), Algebra, vol. Serge Lang ( May 19, 1927 – September 12, 2005) was a French -born American Mathematician. 211, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, MR 1878556, ISBN 978-0-387-95385-4
Silverman, Joeseph (1993). Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many The Arithmetic of Elliptic Curves. Springer. ISBN 0-387-96203-4.

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