Algebraic number theory

What is the discriminant $d_K$ of the quadratic field $K = \mathbb{Q}(\sqrt{-7})$?

Which ring is the ring of integers of $\mathbb{Q}(\sqrt{13})$?

List the primes $p < 25$ that are inert (remain prime) in the Gaussian integers $\mathbb{Z}[i]$. Enter them as a set, for example {2, 5}.

In the proof that a Dedekind UFD is a PID, we find an irreducible $\pi$ with $(\pi) \subseteq \mathfrak{p}$ for a non-zero prime $\mathfrak{p}$ and conclude $(\pi) = \mathfrak{p}$. Which property is used for this last step?

Using the Minkowski bound $M_K \approx 2.85$ for $K = \mathbb{Q}(\sqrt{-5})$, what is the class number $h_K$?

In $R = \mathbb{Z}[\sqrt{-3}]$, let $\mathfrak{p} = (2, 1 + \sqrt{-3})$. One checks that $\mathfrak{p}^2 = 2\mathfrak{p}$ although $\mathfrak{p} \neq (2)$. Which hypothesis of Theorem 4 fails, and where does the proof break?

What is the rank of the unit group $\mathcal{O}_K^\times$ for $K = \mathbb{Q}(\sqrt[3]{2})$? (The minimal polynomial $x^3 - 2$ has one real root and two complex conjugate roots.)

Which pair of Lean facts about ℤ√(-5) (Mathlib's $\mathbb{Z}[\sqrt{-5}]$) is exactly what the lesson uses to prove ¬ UniqueFactorizationMonoid (ℤ√(-5))?

Write inert_primes(d, bound) returning the sorted list of primes $p < \text{bound}$ that are inert in $\mathbb{Q}(\sqrt d)$ ($d \ne 0, 1$ squarefree). Use Dedekind–Kummer: with $\omega = \sqrt d$ ($d \equiv 2, 3 \bmod 4$) or $\omega = \frac{1 + \sqrt d}{2}$ ($d \equiv 1 \bmod 4$), $p$ is inert exactly when the minimal polynomial of $\omega$ has no root modulo $p$.