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
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
Usually we identify K with its image ι(K), and in this case K becomes a subﬁeld
We write L : K for an extension where K is a subﬁeld of L. In this case, ι is the
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.