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By viewing the regular $N$-gon as the set of $N$th roots of unity in thecomplex plane we transform several questions regarding polygon diagonals intowhen a polynomial vanishes when evaluated at roots of unity. To study thesesolutions we implement algorithms in Sage as well as examine a trigonometricdiophantine equation. In doing so we classify when a metallic ratio can berealized as a ratio of polygon diagonals, answering a question raised in a PBSInfinite Series broadcast. We then generalize this idea by examining the degreeof the number field generated by a given ratio of polygon diagonals.

Cyclotomic Points and Algebraic Properties of Polygon Diagonals