Algebraic Number Theory

In the ordinary integers, 5 is prime — indivisible. But step into the Gaussian integers ℤ[i] (where you allow √−1), and 5 splits cleanly: 5 = (2 + i)(2 − i). Neither factor is a unit times 5. The moment you enlarge the number system, 'prime' becomes a relative notion — primes split, stay inert, or ramify — and unique factorization can even break. The rescue, one of the great ideas of 19th-century math, is to factor not numbers but ideals.