5 - The Galois Correspondence

Summary

We are now ready to introduce one of the most elegant results in algebra, the Galois correspondence. This correspondence gives us a framework for understanding the relationships between the structure of a splitting field over a field K and the structure of its group of automorphisms over K. Once we establish this correspondence, we go on to study in some detail how the group provides some distinguishing characteristics of the various conjugates over K of an element of the splitting field.

Normal Field Extensions and Splitting Fields

The property of splitting fields that encapsulates much of the information necessary for proving steps in the Galois correspondence is that of normality. In this section, we introduce the property and explore its connection with splitting fields.

Definition 23.1 (Normal Field Extension in ℂ). Let K be a subfield of ℂ. An algebraic field extension L/K is normal if every polynomial f ∈ K[X] that is irreducible over K and that has at least one root in L contains n = deg(f) roots in L.

Note that since the roots of an irreducible polynomial in K[X] are distinct (Theorem11.3), this definition is equivalent to

Definition 23.2 (Normal Field Extension). An algebraic field extension L/K is normal if every polynomial p ∈ K[X] that is irreducible over K and that has at least one root in L factors into linear terms over L.

Clearly a splitting field of an irreducible polynomial p ∈ K[X] satisfies the condition for the particular polynomial p, but the interest of the property lies in whether the condition is satisfied for all irreducible polynomials q ∈ K[X]: each such polynomial must have either none of its roots in L or all of its roots in L.