field extension as a vector space

Here linear algebra reaches into number theory. If a field K contains a smaller field F (an extension, written K over F), then K is automatically a vector space over F: its elements are the 'vectors', the elements of F are the scalars, and the field's own addition and multiplication supply the vector-space operations. The same set K wears two hats — a field in its own right, and a vector space over the subfield F.

Once K is a vector space over F it has a dimension, and that dimension gets a special name: the degree of the extension, written [K : F]. So the bridge is exact — a question about how much bigger K is than F becomes a question about dimension, answerable with bases and the dimension theorem. The whole machinery of Vol I and this field is suddenly available to study fields.

The cleanest example is C over R. Every complex number is a + b*i with a, b real, so {1, i} is a basis and C is a 2-dimensional real vector space: [C : R] = 2. Likewise Q(sqrt 2), the rationals with sqrt 2 adjoined, has basis {1, sqrt 2} over Q, every element being a + b*sqrt 2 with a, b rational, so its degree is 2. Adjoining a cube root gives degree 3, with basis {1, c, c^2}.

Why this matters: the multiplicativity of degree — if F < K < L then [L : F] = [L : K]*[K : F] — is just the statement that dimensions multiply for towers of spaces, and it is the engine behind classic impossibility proofs (you cannot trisect an angle or double the cube with compass and straightedge). A theorem that sounds like geometry is settled by counting dimensions of field extensions. Linear algebra is doing the heavy lifting in disguise.