Find the minimal polynomial of an arbitrary root of unity [math]w\in\mathbb T[/math].

Develop the theory of the conjecture [math]H_{3n}(3)\neq\emptyset[/math], in analogy with the theory of the Hadamard conjecture, namely [math]H_{4n}(2)\neq\emptyset[/math].

Prove that for any [math]l\in\mathbb N[/math], any vanishing sum of [math]l[/math]-roots of unity appears as a sum of cycles, with [math]\mathbb Z[/math] coefficients.

Prove that for [math]l=p^aq^b[/math], any vanishing sum of [math]l[/math]-roots of unity appears as a sum of cycles.

Read the proof of the Lam-Leung theorem, stating that the lenght of a vanishing sum of roots of unity should equal the length of a sum of cycles, and write down a brief account of that proof, explaining the main ideas there.

Work out all the details for the dichotomy in Proposition 6.18.

Prove that the [math]7\times7[/math] regular matrices can only have

as cycle structure, then prove that the case [math]5+2[/math] is actually excluded.

In the context of the previous exercise, prove that the cases

do not interact, in the sense that a regular [math]7\times7[/math] Hadamard matrix has either all scalar products between the rows of type [math]3+2+2[/math], or of type [math]7[/math].

Prove that the Fourier matrix [math]F_7[/math] is the only [math]7\times7[/math] complex Hadamard matrix having cycle structure [math]7[/math].