Algebraic Integers

Roots of monic integer polynomials — the rings O_K in number fields, lattices in the complex plane

Select a number field to explore its ring of integers
6
5
α algebraic integer ⟺ ∃ monic f ∈ ℤ[x]: f(α)=0 ⟺ α integral over ℤ

In ℚ(√d), the ring of integers is ℤ[√d] if d≢1 mod 4, or ℤ[(1+√d)/2] if d≡1 mod 4. Gaussian integers ℤ[i] have unique factorization; ℤ[√-5] does not: 6 = 2·3 = (1+√-5)(1-√-5). The class group measures the failure of unique factorization.