What do clock arithmetic, polynomials, the complex numbers and the codes in a QR code have in common? They're all rings: number systems where you can add, subtract and multiply. This course builds the theory from the ground up, with pictures first and proofs close behind.
Start with Unit 1Every concept starts with a familiar example and a plain-language explanation of why it matters.
Clocks, grids, lattices and diagrams that let you see the structure before the symbols.
Real theorems with full proofs, tucked into expandable boxes, plus optional "going deeper" sections.
Questions throughout and a practice set at the end of each unit, with answers hidden until you're ready.
Work through the units in order. Each one builds on the ones before it. You need school algebra; everything else is explained as it comes up.
Number systems you already know; the seven axioms; first consequences like \(0 \cdot a = 0\); units and zero divisors.
Start hereCongruences and the rings \(\mathbb{Z}/n\); which elements have inverses; Euclid's algorithm; Fermat's little theorem; RSA.
Available nowWhen cancellation works; why every finite domain is a field; fractions; characteristic; the freshman's dream.
Available nowLong division; roots and factors; polynomials over \(\mathbb{Z}/p\); irreducible polynomials and how to test for them.
Available nowMaps that respect arithmetic; proving equations have no solutions; kernels; isomorphisms; \(\mathbb{Z}/6 \cong \mathbb{Z}/2 \times \mathbb{Z}/3\).
Available nowDeclaring things to be zero; quotient rings; building \(\mathbb{C}\) from real polynomials; the first isomorphism theorem.
Available nowWhich quotients are domains and fields; building new fields from irreducible polynomials; a map of the primes.
Available nowIrreducibles versus primes; how unique factorisation fails in \(\mathbb{Z}[\sqrt{-5}]\); Euclidean domains, PIDs and UFDs; ideal numbers.
Available nowPrimes in the plane; Fermat's theorem on sums of two squares; counting lattice points; Pythagorean triples.
Available nowWhy sizes are prime powers; \(\mathbb{F}_4\), \(\mathbb{F}_8\) and \(\mathbb{F}_9\); generators; one field of each size; error-correcting codes and cryptography.
Available nowThe Chinese remainder theorem; Noetherian rings; ideals as geometric shapes; quaternions; where to go next.
Available now