Separable Extensions, An Introduction

Introduction

If K is a field and p is an irreducible polynomial over K, there is some splitting field F/K that contains all the roots of p. In fact the splitting field can be constructed by adjoining the roots of p in succession. See splitting fields for more details.

Recall that the roots of p(x) are called conjugates. We showed that these conjugates are indistinguishable. There is no way to tell one from another. Like special relativity, there is no "correct" frame of reference. As a corollary, all the roots of p(x) have a common multiplicity. If there are 7 instances of u in p(x) there are 7 instances of v in p(x), and so on.

The irreducible polynomial p(x) over K is separable if its roots, in the splitting field F/K, are all distinct. In other words, the common multiplicity is 1. Each root appears once, and there are n different roots, all conjugate, and all indistinguishable.

The algebraic element u in F/K is separable if the irreducible polynomial associated with u is separable.

The extension F/K is separable if every u in F is separable. Therefore every separable extension is algebraic.

The field K is perfect if every algebraic extension of K is separable. We will show that Q and Z_p are perfect, which is why most fields are separable.

An inseparable polynomial is irreducible, but not separable. An inseparable element u in F/K is algebraic, but not separable. An inseparable extension is algebraic, but not separable.

You can easily study algebraic extensions for a lifetime and never run into an inseparable extension. They are quite rare. But you'll find some here.