ABSTRACT
In this chapter we define the important concepts of normality and separability and develop some key properties.
Suppose that K is a subfield of Often a polynomial has no zeros in K. But it must have zeros in by the Fundmental Theorem of Algebra. Therefore, it may have some zeros, at least, in some given extension field L of K. For example, has no zeros in but has zeros We shall study this phenomenon in detail, showing that every polynomial can be resolved into a product of linear factors (and hence has its full complement of zeros) if the ground field K is extended to a suitable splitting field N. An extension N:K is normal if any irreducible polynomial over K with at least one zero in N splits into linear factors in N. We show that an extension is normal if and only if it is a splitting field.
