Page 268 - DMTH403_ABSTRACT_ALGEBRA

Unit 27: Separable Extensions

5. If D : R  R is a derivatives and f  F0 is the corresponding derivation on R[x] from its ring Notes
of constants in C[x], where ................... is the constant for ..................., f(x) =
1
 a x in R[x] and fd(x)   D(a )X i
1
i
u
(a) D, C (b) C, D
(c) X, C (d) C, D

27.2 Summary

Let L/K be a finite extension. Then L is separable over K if and only if any derivation of K

has a unique extension to a derivation of L.
If L = K(a ,....., a ) and each a is separable over K then every element of L is separable over
 1 r i
K (so L/K is separable).
Let L/K be a finite extension and F be an intermediate field. If L/F and F/K are separable

then L/K is separable.
The proof of Theorem 2 implies Theorem 4.

Let L/K be an extension of fields, and   L be algebraic over K. Then is separable over K

if and only if any derivation on K has a unique extension to a derivation on K().
A commutative ring with no nonzero nilpotent elements is called reduced.

A domain is reduced, but a more worthwhile example is a product of domains, like

F3 × Q[X], which is not a domain but is reduced.
An arbitrary field extension L/K is called separable when the ring K  L is reduced.
 K
A derivation is an abstraction of differentiation on polynomials. We want to work with

derivations on fields, but polynomial rings will intervene, so we need to understand
derivations on rings before we focus on fields.
27.3 Keywords

Separability: Separability of a finite field extension L/K can be described in several different
ways.

Commutative Ring: A commutative ring with no nonzero nilpotent elements is called reduced.
Domain: A domain is reduced, but a more worthwhile example is a product of domains, like
F3 × Q[X], which is not a domain but is reduced.

Derivation: A derivation is an abstraction of differentiation on polynomials.
27.4 Review Questions

1. Let R be a domain with fraction field K. Any derivation D: R  K uniquely extends to
D : K  K, given by the quotient rule: D (a/b) = (bD(a) – aD(b))/b . Prove it.
2


2. Let L/K be a finite extension of fields, and D: K  K be a derivation. Suppose
a  L is separable over K, with minimal polynomial (X)  K[X]. That is, (X) is irreducible
in K[X], () = 0, and ’()  0. Then D has a unique extension from K to a derivation on the
field K(), and it is given by the rule.

LOVELY PROFESSIONAL UNIVERSITY 261