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Sachin Kaushal, Lovely Professional University Unit 27: Separable Extensions

Unit 27: Separable Extensions Notes

CONTENTS
Objectives

Introduction
27.1 Separability
27.2 Summary
27.3 Keywords
27.4 Review Questions

27.5 Further Readings

Objectives

After studying this unit, you will be able to:
Define separability

Discuss examples related to separable extension

Introduction

In the last unit, you have studied about the splitting field and extension field. This unit will
provide you information related to separable extension.

27.1 Separability

Separability of a finite field extension L/K can be described in several different ways. The
original definition is that every element of L is separable over K (that is, has a separable
minimal polynomial in K[X]). We will give here three descriptions of separability for a finite
extension and use each of them to prove two theorems about separable extensions.
Theorem 1: Let L/K be a finite extension. Then L/K is separable if and only if the trace function
Tr L/K : L  K is not identically 0.
The trace function is discussed in Appendix A.

Theorem 2: Let L/K be a finite extension. Then L/K is separable if and only if the ring K  K L
has no non-zero nilpotent elements. When L/K is separable, the ring K  K L is isomorphic to

K [L:K] .

Example: Consider the extension Q( 2 )=Q. Since Q( 2 )  Q[X]/(X – 2), tensoring with
2
2
2
Q gives Q  Q Q   ; Q[X]/(X  2)  Q[X]/((X  2)(X  2)  Q Q,

which is a product of 2 copies of Q (associated to the 2 roots of X 2) and has no nilpotent
2
elements besides 0.

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