1.5. Extension fields; degrees.

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 A field E containing a field F is called an extension (field)

of F. Such an E can be regarded (in an obvious fashion) as an F-vector space. We write

[E : F] for the dimension (possibly infinite) of E as an F-vector space, and call [E : F] the

degree of E over F. We often say that E is finite over F when it has finite degree over F.

Example 1.9. (a) The field of complex numbers C has degree 2 over R (basis {1, i}).

(b) The field of real numbers R has infinite degree over Q. (We know Q is countable,

which implies that any finite-dimensional vector space over Q is countable;but R is not

countable. More explicitly, one can find real numbers α such that 1, α, α2, . . . are linearly

independent (see section 1.9 below)).

(c) The field of Gaussian numbers Q(i) =df {a + bi ∈ C | a, b ∈ Q} has degree 2 over Q

(basis {1, i}).

(d) The field F(X) has infinite degree over F. (It contains the F-subspace F[X], which

has the infinite basis {1,X,X2, . . . }.)

Proposition 1.10. Let L ⊃ E ⊃ F (all fields). Then L/F is of finite degree ⇐⇒ L/E

and E/F are both of finite degree, in which case

[L : F] = [L : E][E : F].

Proof. Assume that L/E and E/F are of finite degree, and let {ei} be a basis for E/F

and {_j} a basis for L/E. I claim that {ei_j} is a basis for L over F. I first show that it

spans L. Let γ ∈ L. Then, because {_j} spans L as an E-vector space,

γ =

_

αj_j , some αj ∈ E,

and because {ei} spans E as an F-vector space, for each j,

αj =

_

aijei, some aij ∈ F.

On putting these together, we find that

γ =

_

aijei_j .

Next I show that {ei_j} is linearly independent. A linear relation

_

aijei_j = 0 can be

rewritten

_

j(

_

i aijei)_j = 0. The linear independence of the _j ’s now shows that

_

i aijei =

0 for each j, and the linear independence of the ei’s now shows that each aij = 0.

Conversely, if L is of finite degree over F, then it is certainly of finite degree over E.

Moreover, E, being a subspace of a finite dimensional F-space, is also finite dimensional.

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