
Chapter 9
Normality and Separability
In this chapter we define the important concepts of normality and separability for
field extensions, and develop some of their key properties.
Suppose that K is a subfield of C. Often a polynomial p(t) ∈ K[t] has no zeros
in K. But it must have zeros in C, by the Fundamental Theorem of Algebra, Theo-
rem 2.4. Therefore it may have at least some zeros in a given extension field L of K.
For example t
2
+ 1 ∈ R[t] has no zeros in R, but it has zeros ±i ∈ C, in Q(i), and for
that matter in any subfield containing Q(i). We shall study this phenomenon in detail,
showing that every polynomial can be resolved into a product of