$2^n$-th rational reciprocity laws
Solution 1:
For $16$th powers see Leonard and Williams. For a general law in function fields, see Hamahata. Patterns for $n \le 3$ are misleading since ${\mathbb Q}(\zeta_8)$ is a biquadratic number field, whereas ${\mathbb Q}(\zeta_{2^n})$ for $n \ge 4$ is not.