The theme of this chapter is the construction of automorphisms to given speci-
ﬁcations. We begin with a generalisation of a K-automorphism, known as a K-
monomorphism. For normal extensions we shall use K-monomorphisms to build up
K-automorphisms. Using this technique, we can calculate the order of the Galois
group of any ﬁnite normal extension, which combines with the result of Chapter 10
to give a crucial part of the fundamental theorem of Chapter 12.
We also introduce the concept of a normal closure of a ﬁnite extension. This
useful device enables us to steer around some of the technical obstructions caused by
We begin by generalising the concept of a K-automorphism of a subﬁeld L of C,
by relaxing the condition that the map should be onto. We continue to require it to be
Deﬁnition 11.1. Suppose that K is a subﬁeld of each of the subﬁelds M and L of C.
Then a K-monomorphism of M into L is a ﬁeld monomorphism φ : M → L such that
φ(k) = k for every k ∈ K.
Example 11.2. Suppose that K = Q,M = Q(α) where α is a real cube root of 2, and
L = C. We can deﬁne a K-monomorphism φ : M → L by insisting that φ(α) = ωα,
where ω = e
. In more detail, every element of M is of the form p + qα + rα
where p,q,r ∈ Q, and
φ(p + qα + rα
) = p + qωα + rω
Since α and ωα have the same minimal polynomial, namely t
−2, Corollary 5.13
implies that φ is a K-monomorphism.
There are two other K-monomorphisms M → L in this case. One is the identity,
and the other takes α to ω
α (see Figure 18).
In general if K ⊆ M ⊆ L then any K-automorphism of L restricts to a K-
monomorphism M → L. We are particularly interested in when this process can be
146 Field Automorphisms
FIGURE 18: Images of Q-monomorphisms of α = Q(
2) : Q.
Theorem 11.3. Suppose that L : K is a ﬁnite normal extension and K ⊆ M ⊆ L. Let
τ be any K-monomorphism M → L. Then there exists a K-automorphism σ of L such
Proof. By Theorem 9.9, L is the splitting ﬁeld over K of some polynomial f over K.
Hence it is simultaneously the splitting ﬁeld over M for f and over τ(M) for τ( f ).
is the identity, so τ( f ) = f . We have the diagram
M → L
τ ↓ ↓ σ
τ(M) → L
with σ yet to be found. By Theorem 9.6, there is an isomorphism σ : L → L such that
= τ. Therefore σ is an automorphism of L, and since σ |
is the identity,
σ is a K-automorphism of L.
This result can be used to construct K-automorphisms:
Proposition 11.4. Suppose that L : K is a ﬁnite normal extension, and α,β are zeros
in L of the irreducible polynomial p over K. Then there exists a K-automorphism σ
of L such that σ(α) = β .
Proof. By Corollary 5.13 there is an isomorphism τ : K(α) → K(β ) such that τ|
is the identity and τ(α) = β . By Theorem 11.3, τ extends to a K-automorphism σ
11.2 Normal Closures
When extensions are not normal, we can try to recover normality by making the
Deﬁnition 11.5. Let L be a ﬁnite extension of K. A normal closure of L : K is an
extension N of L such that