The Langlands programme: an orientation

Use quadratic reciprocity to compute the Legendre symbol $\left(\frac{3}{11}\right)$. Enter $1$ or $-1$.

Into how many distinct prime ideals does $p = 7$ split in the cyclotomic field $\Q(\zeta_{12})$, which has degree $\varphi(12) = 4$?

Let $g = \sum_{a=1}^{6} \left(\frac{a}{7}\right)\zeta_7^a$ be the quadratic Gauss sum for $q = 7$. What is the integer $g^2$?

In the Galois-theoretic proof of quadratic reciprocity we used that $\Q(\sqrt{q^*})$ is the unique quadratic subfield of $\Q(\zeta_q)$. Why is that true?

Which statement about the number of roots of $x^3 - x - 1$ modulo a prime $p \neq 23$ is true?

By Chebotarev's density theorem, what proportion of primes $p$ have the property that $x^3 - x - 1$ has exactly one root mod $p$?

Which of the lesson's Lean theorems is the formal version of the statement "the Frobenius element at $p$ in $\operatorname{Gal}(\Q(\zeta_n)/\Q) \cong (\Z/n)^\times$ is the class of $p$" used in the proof of quadratic reciprocity?

Write tau_list(N) returning the list [0, τ(1), …, τ(N-1)] of coefficients of $\Delta = q\prod_{n \ge 1}(1 - q^n)^{24}$, so that tau_list(N)[n] is $\tau(n)$. Then test Mordell's theorem $\tau(mn) = \tau(m)\tau(n)$ for coprime $m, n$ with $mn \lt 60$.

Which of the following are proved theorems? Choose all that apply.