discriminant of a number field
A number field is, geometrically, a lattice of integers sitting inside a real or complex space, and like any lattice it has a notion of volume and of how “stretched” it is. The discriminant is a single integer that packages this geometric data and, remarkably, simultaneously records exactly which primes ramify. It is the number field's arithmetic fingerprint.
Let O_K have an integral basis b_1, ..., b_n and let s_1, ..., s_n be the n embeddings of K into C. The discriminant is d_K = det([s_j(b_i)])^2, the squared determinant of the matrix of embeddings applied to the basis. It is an integer independent of the chosen basis, and equals the determinant of the trace form matrix [Tr(b_i b_j)]. Its absolute value is the squared covolume of O_K as a lattice.
The discriminant's prime divisors are exactly the primes that ramify in K (the Dedekind discriminant theorem), so a prime ramifies if and only if it divides d_K. Minkowski's theorem forces |d_K| to grow with the degree, implying Q is the only number field with discriminant of absolute value 1 and that there are only finitely many number fields with bounded discriminant.
For Q(sqrt(d)) with d squarefree, the discriminant is d if d ≡ 1 (mod 4) and 4d otherwise. So Q(sqrt(5)) has discriminant 5, while Q(sqrt(2)) has discriminant 8 and Q(sqrt(-1)) has discriminant -4.
That 2 divides the discriminant of Q(sqrt(2)) and Q(sqrt(-1)) but not Q(sqrt(5)) reflects exactly that 2 ramifies in the first two but not the third.