A field is a place where you can add, subtract, multiply and — crucially — divide by anything nonzero: \mathbb{Q}, \mathbb{R}, \mathbb{C}, the integers mod a prime. But fields rarely live alone. The rationals sit inside the reals; the reals sit inside the complex numbers. Each time, a small field is wrapped inside a larger one that obeys the very same arithmetic. That nesting is a field extension, and it is the single idea this whole module is built on.

Write E/F (read "E over F") when F is a subfield of E — every element of F is an element of E, and F uses the addition and multiplication it inherits from E. So \mathbb{C}/\mathbb{R} and \mathbb{R}/\mathbb{Q} are extensions. The slash is not a quotient — it is just the traditional notation for "the bigger field, sitting over the smaller one." We call F the base field (or ground field).

The revelation that unlocks everything is a change of viewpoint. Forget for a moment that E is a field. Just multiply its elements by scalars taken from F. Suddenly E is a vector space over F — and all the machinery of linear algebra, basis and dimension included, comes flooding in to measure "how much bigger" E is than F.

The founding example: complex over real

You have known your first field extension since you learned about complex numbers. Every complex number is a + bi with a, b \in \mathbb{R}. Read that as a scalar combination: a \cdot 1 + b \cdot i. The complex field \mathbb{C} is a vector space over \mathbb{R} with basis \{1, i\} — it is two-dimensional. That single number, 2, is the reason the complex plane is a plane.

The same story runs with \mathbb{Q}(\sqrt 2), the smallest field containing both \mathbb{Q} and \sqrt 2. Its elements are exactly a + b\sqrt 2 with a, b \in \mathbb{Q} — a two-dimensional vector space over \mathbb{Q} with basis \{1, \sqrt 2\}. Adding is obvious; the only thing to check is that dividing stays inside, and the old trick of rationalising the denominator does it:

which is again of the form c + d\sqrt 2. (The denominator a^2 - 2b^2 is never zero for rational a, b not both zero, precisely because \sqrt 2 is irrational.) So \mathbb{Q}(\sqrt 2) really is a field, and \mathbb{Q}(\sqrt 2)/\mathbb{Q} is an extension.

Building extensions by adjoining

Where do extensions come from? You adjoin a new element. Given a field F living inside a bigger field, and an element \alpha of that bigger field, write F(\alpha) for the smallest field that contains all of F and \alpha too. It must contain \alpha^2, \alpha^3, \dots, sums like 3 + 2\alpha, and quotients of such things — everything you are forced to include to keep the field axioms. Formally it is the field of rational expressions in \alpha with coefficients in F.

So \mathbb{R}(i) = \mathbb{C} and \mathbb{Q}(\sqrt 2) is as above. You can adjoin several elements: \mathbb{Q}(\sqrt 2, \sqrt 3) is the smallest field holding both roots, and it turns out to be four-dimensional over \mathbb{Q}, with basis \{1, \sqrt 2, \sqrt 3, \sqrt 6\}. An extension you can reach by adjoining a single element, E = F(\alpha), is called simple — and a surprising theorem later in the module says almost every extension you meet is secretly simple.