Hermite–Minkowski theorem

Remarks

Part 1 follows from the Minkowski bound: every ideal class contains a nonzero integral ideal of norm at least $1$ and at most $M_K$, so $M_K \ge 1$. Since $2 r_2 \le n$ and $\pi/4 < 1$, the right side is at least $(\pi/4)^n (n^n/n!)^2$, which is $\pi^2/4 > 2$ for $n = 2$ and increases with $n$. A prime $p$ ramifies in $K$ exactly when $p$ divides $d_K$, so $\abs{d_K} > 1$ gives a ramified prime.

For part 2, the bound of part 1 first bounds the degree $n$. Then Minkowski's theorem on convex bodies gives a primitive element of $K$ whose conjugates are bounded in terms of $D$. So its minimal polynomial has bounded integer coefficients, and there are finitely many such polynomials.

Example: $\Q(\sqrt{-3})$ has $\abs{d_K} = 3$, so the bound $\abs{d_K} \ge 3$ for $n = 2$ is sharp.

Going deeper

For a student

Part 1 says that $\Q$ has no nontrivial extension that is unramified at all primes. Many other number fields do have such extensions, and the abelian ones are described by the Hilbert class field. Part 2 makes tables of number fields ordered by discriminant possible, since each range contains finitely many fields.