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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