From a field to a bigger field
A field is a number system where you can add, subtract, multiply, and divide (by anything except 0) and the usual rules hold. The rational numbers Q form a field; so do the reals R. A field extension is just a field K that contains a smaller field F as a sub-system. We write it K/F (read “K over F”), and F is called the base field.
The smallest interesting example: start with Q and throw in √2. To stay closed under +, −, ×, ÷ you are forced to include every number of the form a + b√2 with a, b rational. That collection is written Q(√2), and it really is a field — you can even divide.
Dividing inside Q(√2) — rationalize the denominator: 1 / (1 + √2) = (1 − √2) / ((1 + √2)(1 − √2)) multiply by the conjugate = (1 − √2) / (1 − 2) = (1 − √2) / (−1) = −1 + √2 Result is again of the form a + b√2 (a = −1, b = 1). Closed!
An extension is a vector space
Here is the key shift in viewpoint. Treat the big field K as a vector space over the small field F: vectors are elements of K, scalars come from F. In Q(√2), every element a + b√2 is an F-combination of the two “vectors” 1 and √2. So {1, √2} is a basis, and the dimension is 2.
That dimension has a name: the degree of the extension, written [K : F]. So [Q(√2) : Q] = 2. The degree is a single number that captures “how much bigger” the extension is — and it will turn out to be the deepest measuring stick in the whole theory.
Algebraic vs. transcendental
Why was Q(√2) only 2-dimensional and not infinite? Because √2 satisfies a polynomial equation with rational coefficients: x² − 2 = 0. A number that is the root of some such polynomial is called algebraic over F. Adjoining one algebraic number always gives a finite-degree extension.
Some numbers are not roots of any rational polynomial at all — these are transcendental, like π and e. Adjoining one of those gives an infinite-dimensional extension. For Galois theory we will stay almost entirely in the friendly, finite, algebraic world.