degree of extension

Definition

The degree of a field extension is

is called finite if and infinite otherwise.

Examples

• For , by cardinality of versus .

• For , since a basis for over is .

Transitivity of degree

Let and be extensions. Then is finite if and only if and are finite, and in this case

Proof

Assume is finite. Then as a vector space over is finite dimensional. From this, is a vector subspace then is finite (subspaces have same dimension or less).

Thinking of as a vector space, assume for contradiction that is infinite dimensional. Then is infinite set of linear independent vectors. But is finite dimensional (as a vector space over ) so there exists a linear relationship

so the let cannot be linear independent.

Assume now that and are finite. Since they are finite dimensional subspaces, we can choose basis and respectively.

Take the candidate basis .

is spanning: An arbitrary element can be written as

Now each element which has basis so

which can be written as

is linearly independent: Consider

Then

So for all , but is a basis, so .