A field is a set with two operations — addition and multiplication — that behave the way you’d expect: you can add, subtract, multiply, and divide (except by zero), and the usual rules (commutativity, associativity, distributivity) hold.

Familiar fields

• $\mathbb{Q}$ — the rationals (fractions $p/q$). The most natural home for algebra.
• $\mathbb{R}$ — the reals. Fill in the gaps (like $\sqrt{2}$, $\pi$).
• $\mathbb{C}$ — the complex numbers $a + bi$. Every polynomial has roots here.
• $\mathbb{F}_p$ — finite fields (integers mod a prime $p$). Only $p$ elements, but fully a field.

Field extensions

The key idea: sometimes you need to enlarge a field to express something. The equation $x^2 = 2$ has no solution in $\mathbb{Q}$, but it does in a bigger field.

Adjoining an element: $\mathbb{Q}(\sqrt{2})$ means “the rationals, plus $\sqrt{2}$, plus everything you can build from those using $+, -, \times, \div$.” Every element of $\mathbb{Q}(\sqrt{2})$ can be written as $a + b\sqrt{2}$ where $a, b \in \mathbb{Q}$.

Tower of extensions

You can extend in stages:

$$\mathbb{Q} \subset \mathbb{Q}(\sqrt{2}) \subset \mathbb{Q}(\sqrt{2}, \sqrt{3}) \subset \cdots$$

Each step adds new elements. The degree of an extension $F \subset E$ is the dimension of $E$ as a vector space over $F$: - $[\mathbb{Q}(\sqrt{2}) : \mathbb{Q}] = 2$ (basis: ${1, \sqrt{2}}$) - $[\mathbb{Q}(\sqrt{2}, \sqrt{3}) : \mathbb{Q}] = 4$ (basis: ${1, \sqrt{2}, \sqrt{3}, \sqrt{6}}$)

Degrees multiply in towers: $[E : F] = [E : K] \cdot [K : F]$.

Radical extensions

A radical extension is one built by adjoining roots: $\sqrt[n]{a}$. The quadratic formula involves $\sqrt{b^2 - 4c}$ — one radical. The cubic formula involves nested square roots and cube roots. The quartic formula is even more complicated but still uses only radicals.

A polynomial is solvable by radicals if its roots lie in some radical extension of $\mathbb{Q}$. Galois’s great insight was connecting this algebraic question to a question about symmetry groups.

Why fields matter

• Polynomials live over fields — the coefficients are in $\mathbb{Q}$, the roots might be in a bigger field
• Galois theory studies the relationship between field extensions and symmetry groups
• Finite fields $\mathbb{F}_p$ are where elliptic curve point counting happens (crucial for L-functions and the modularity theorem)
• The failure of unique factorisation in certain field extensions (like $\mathbb{Z}[\sqrt{-5}]$) is what makes Fermat’s Last Theorem hard