Chapter 4

Field Extensions

Galois’s original theory was couched in terms of polynomials over the complex ﬁeld.

The modern approach is a consequence of the methods used, starting around 1890

and ﬂourishing in the 1920s and 1930s, to generalise the theory to arbitrary ﬁelds.

From this viewpoint the central object of study ceases to be a polynomial, and be-

comes instead a ‘ﬁeld extension’ related to a polynomial. Every polynomial f over a

ﬁeld K deﬁnes another ﬁeld L containing K (or at any rate a subﬁeld isomorphic to

K). There are conceptual advantages in setting up the theory from this point of view.

In this chapter we deﬁne ﬁeld extensions (always working inside C) and explain the

link with polynomials.

4.1 Field Extensions

Suppose that we wish to study the quartic polynomial

f (t) = t

4

−4t

2

−5

over Q. Its irreducible factorisation over Q is

f (t) = (t

2

+ 1)(t

2

−5)

so the zeros of f in C are ±i and ±

√

5. There is a natural subﬁeld L of C associated

with these zeros; in fact, it is the unique smallest subﬁeld that contains them. We

claim that L consists of all complex numbers of the form

p + qi + r

√

5 + si

√

5 (p, q, r,s ∈ Q)

Clearly L must contain every such element, and it is not hard to see that sums and

products of such elements have the same form. It is harder to see that inverses of

(non-zero) such elements also have the same form, but it is true: we postpone the

proof to Example 4.8. Thus the study of a polynomial over Q leads us to consider a

subﬁeld L of C that contains Q. In the same way the study of a polynomial over an

arbitrary subﬁeld K of C will lead to a subﬁeld L of C that contains K. We shall call

L an ‘extension’ of K. For technical reasons this deﬁnition is too restrictive; we wish

to allow cases where L contains a subﬁeld isomorphic to K, but not necessarily equal

to it.

63

64 Field Extensions

Deﬁnition 4.1. A ﬁeld extension is a monomorphism ι : K → L, where K and L are

subﬁelds of C. We say that K is the small ﬁeld and L is the large ﬁeld.

Notice that with a strict set-theoretic deﬁnition of function, the map ι determines

both K and L. See Deﬁnition 1.3 for the deﬁnition of ‘monomorphism’. We often

think of a ﬁeld extension as being a pair of ﬁelds (K,L), when it is clear which

monomorphism is intended.

Examples 4.2. 1. The inclusion maps ι

1

: Q →R,ι

2

: R →C, and ι

3

: Q →C are all

ﬁeld extensions.

2. Let K be the set of all real numbers of the form p + q

√

2, where p, q ∈Q. Then K

is a subﬁeld of C by Example 1.7. The inclusion map ι : Q → K is a ﬁeld extension.

If ι : K → L is a ﬁeld extension, then we can usually identify K with its image

ι(K), so that ι can be thought of as an inclusion map and K can be thought of as a

subﬁeld of L. Under these circumstances we use the notation

L : K

for the extension, and say that L is an extension of K. In future we shall identify K

and ι(K) whenever this is legitimate.

The next concept is one which pervades much of abstract algebra:

Deﬁnition 4.3. Let X be a subset of C. Then the subﬁeld of C generated by X is the

intersection of all subﬁelds of C that contain X.

It is easy to see that this deﬁnition is equivalent to either of the following:

1. The (unique) smallest subﬁeld of C that contains X.

2. The set of all elements of C that can be obtained from elements of X by a ﬁnite

sequence of ﬁeld operations, provided X 6= {0} or /0.

Proposition 4.4. Every subﬁeld of C contains Q.

Proof. Let K ⊆ C be a subﬁeld. Then 0,1 ∈ K by deﬁnition, so inductively we ﬁnd

that 1 + ... + 1 = n lies in K for every integer n > 0. Now K is closed under additive

inverses, so −n also lies in K, proving that Z ⊆ K. Finally, if p,q ∈ Z and q 6= 0,

closure under products and multiplicative inverses shows that pq

−1

∈ K. Therefore

Q ⊆ K as claimed.

Corollary 4.5. Let X be a subset of C. Then the subﬁeld of C generated by X con-

tains Q.

Because of Corollary 4.5, we use the notation

Q(X)

for the subﬁeld of C generated by X.

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