Let K be a field. The next theorem gives a connection between linear algebra (see the prerequisites) and elements of a field extension.
Theorem
Suppose that L is a subfield of the field K.
- Then K is a vector space over L.
- Moreover,
for each x
K,
multiplication with x is a linear
transformation of this vector space over L.
Below one finds a consequence for finite fields: their orders form a proper subset of the natural numbers.
Corollary
If F is a finite field, then there is a prime p and a natural number n such that |F| = pn.