Chapter 17
Abstract Field Extensions
Having defined rings and fields, and equipped ourselves with several methods for
constructing them, we are now in a position to attack the general structure of an
abstract field extension. Our previous work with subfields of C paves the way, and
most of the effort goes into making minor changes to terminology and checking
carefully that the underlying ideas generalise in the obvious manner.
We begin by extending the classification of simple extensions to general fields.
Having done that, we assure ourselves that the theory of normal extensions, including
their relation to splitting fields, carries over to the general case. A new issue, sepa-
rability, comes into play when the characteristic of the field is not zero. The main
result is that the Galois correspondence can be set up for any finite separable normal
extension, and it then has exactly the same properties that we have already proved
over C.
Convention on Generalisations. Much of this chapter consists of routine veri-
fication that theorems previously stated and proved for subfields or subrings of C
remain valid for general rings and fields—and have essentially the same proofs. As
a standing convention, we refer to ‘Lemma X.Y (generalised)’ to mean the generali-
sation to an arbitrary ring or field of Lemma X.Y; usually we do not restate Lemma
X.Y in its new form. In cases where the proof requires a new method, or extra hy-
potheses, we will be more specific. Moreover, some of the most important theorems
will be restated explicitly.
17.1 Minimal Polynomials
Definition 17.1. A field extension is a monomorphism ι : K L, where K,L are
fields.
Usually we identify K with its image ι(K), and in this case K becomes a subfield
of L.
We write L : K for an extension where K is a subfield of L. In this case, ι is the
inclusion map.
We define the degree [L : K] of an extension L : K exactly as in Chapter 6. Namely,
consider L as a vector space over K and take its dimension. The Tower Law remains
valid and has exactly the same proof.
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