Chapter 11

Field Automorphisms

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

non-normal extensions.

11.1 K-Monomorphisms

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

one-to-one.

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

2πi/3

. In more detail, every element of M is of the form p + qα + rα

2

where p,q,r ∈ Q, and

φ(p + qα + rα

2

) = p + qωα + rω

2

α

2

Since α and ωα have the same minimal polynomial, namely t

3

−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

2

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

reversed.

145

146 Field Automorphisms

FIGURE 18: Images of Q-monomorphisms of α = Q(

3

√

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

that σ|

M

= τ.

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 ).

But τ|

K

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

σ|

M

= τ. Therefore σ is an automorphism of L, and since σ |

K

= τ|

K

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|

K

is the identity and τ(α) = β . By Theorem 11.3, τ extends to a K-automorphism σ

of L.

11.2 Normal Closures

When extensions are not normal, we can try to recover normality by making the

extensions larger.

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

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