Constructing Regular Polygons -- Primitive Roots of Unity

I have been working for quite a while on finding closed, simple expressions for primitive roots of unity, that is finding solutions of cyclotomic polynomials by radical expressions in the case of constructible regular polygons.

The long-term goal is to find solutions for the 5th, 17th, 257th, and 65537th root of unity, and to better understand relationship between coefficients.

Below some roots already computed, expressed as minimal, regular nested expressions.

This is an onging project.

 Primitive 5th Root of Unity $$\sqrt[5]{1}$$ \style{text-align:right;}{ \begin{split} -\frac{1}{4}& \ \\[10pt]+\frac{1}{4}& \ \sqrt{5\,}\\[10pt]+\frac{1}{4}& \ \sqrt{\!-\!10\!-\!2\sqrt{5\,}\,}\\ \end{split} } Primitive 17th Root of Unity $$\sqrt[17]{1}$$ \style{text-align:right;}{ \begin{split} -\frac{1}{16}& \ \\[10pt]+\frac{1}{16}& \ \sqrt{17\,}\\[10pt]+\frac{1}{16}& \ \sqrt{34\!-\!2\sqrt{17\,}\,}\\[10pt]+\frac{1}{8}& \ \sqrt{17\!+\!3\sqrt{17\,}\!-\!\sqrt{170\!+\!38\sqrt{17\,}\,}\,}\\[10pt]+\frac{1}{8}& \ \sqrt{\!-\!34\!+\!2\sqrt{17\,}\!-\!2\sqrt{34\!-\!2\sqrt{17\,}\,}\!+\!4\sqrt{17\!+\!3\sqrt{17\,}\!+\!\sqrt{170\!+\!38\sqrt{17\,}\,}\,}\,}\\ \end{split} } Primitive 257th Root of Unity $$\sqrt[257]{1}$$ \style{text-align:right;}{ \begin{split} -\frac{1}{256}& \ \\[10pt]+\frac{1}{256}& \ \sqrt{257\,}\\[10pt]+\frac{1}{256}& \ \sqrt{514\!-\!2\sqrt{257\,}\,}\\[10pt]+\frac{1}{128}& \ \sqrt{257\!+\!15\sqrt{257\,}\!+\!\sqrt{58082\!+\!3614\sqrt{257\,}\,}\,}\\[10pt]+\frac{1}{128}& \ \sqrt{514\!-\!18\sqrt{257\,}\!+\!6\sqrt{514\!-\!2\sqrt{257\,}\,}\!+\!4\sqrt{12593\!-\!561\sqrt{257\,}\!+\!\sqrt{21794114\!-\!552002\sqrt{257\,}\,}\,}\,}\\[10pt]+\frac{1}{64}& \ \sqrt{257\!-\!\sqrt{257\,}\!-\!\sqrt{12850\!-\!754\sqrt{257\,}\,}\!+\!2\sqrt{3341\!-\!109\sqrt{257\,}\!-\!\sqrt{6509810\!-\!268274\sqrt{257\,}\,}\,}\!-\!2\sqrt{10794\!+\!70\sqrt{257\,}\!-\!2\sqrt{16703458\!-\!380386\sqrt{257\,}\,}\!-\!4\sqrt{4535793\!+\!253711\sqrt{257\,}\!-\!\sqrt{36803719442626\!+\!2282091196478\sqrt{257\,}\,}\,}\,}\,}\\[10pt]+\frac{1}{64}& \ \sqrt{\begin{aligned}514& \ \!+\!14\sqrt{257\,}\!+\!6\sqrt{514\!-\!2\sqrt{257\,}\,}\!+\!4\sqrt{4369\!+\!15\sqrt{257\,}\!+\!\sqrt{9112706\!-\!354178\sqrt{257\,}\,}\,}\!-\!4\sqrt{4626\!+\!286\sqrt{257\,}\!+\!10\sqrt{247234\!+\!15422\sqrt{257\,}\,}\!+\!4\sqrt{3314529\!+\!206751\sqrt{257\,}\!+\!\sqrt{20768537086978\!+\!1295505749246\sqrt{257\,}\,}\,}\,}\\[-10pt]& \ \!-\!8\sqrt{3341\!+\!139\sqrt{257\,}\!+\!35\sqrt{514\!-\!2\sqrt{257\,}\,}\!+\!2\sqrt{1306845\!+\!53283\sqrt{257\,}\!-\!\sqrt{257924604274\!+\!4074182542\sqrt{257\,}\,}\,}\!-\!2\sqrt{3682810\!+\!172118\sqrt{257\,}\!+\!10\sqrt{12461428850\!+\!278588558\sqrt{257\,}\,}\!+\!4\sqrt{1185667505937\!+\!67520656239\sqrt{257\,}\!+\!\sqrt{375570632850308163054146\!+\!22763424315321766000318\sqrt{257\,}\,}\,}\,}\,}\end{aligned}\,}\\[10pt]+\frac{1}{32}& \ \sqrt{\begin{aligned}257& \ \!-\!\sqrt{257\,}\!-\!\sqrt{514\!-\!2\sqrt{257\,}\,}\!-\!2\sqrt{257\!+\!15\sqrt{257\,}\!+\!\sqrt{58082\!+\!3614\sqrt{257\,}\,}\,}\!+\!6\sqrt{514\!-\!18\sqrt{257\,}\!+\!6\sqrt{514\!-\!2\sqrt{257\,}\,}\!+\!4\sqrt{12593\!-\!561\sqrt{257\,}\!+\!\sqrt{21794114\!-\!552002\sqrt{257\,}\,}\,}\,}\\[-10pt]& \ \!-\!4\sqrt{1285\!-\!5\sqrt{257\,}\!-\!5\sqrt{12850\!-\!754\sqrt{257\,}\,}\!+\!10\sqrt{3341\!-\!109\sqrt{257\,}\!-\!\sqrt{6509810\!-\!268274\sqrt{257\,}\,}\,}\!+\!2\sqrt{442554\!-\!13642\sqrt{257\,}\!+\!6\sqrt{4595699186\!-\!281900018\sqrt{257\,}\,}\!+\!4\sqrt{18581190721\!-\!1150035777\sqrt{257\,}\!+\!\sqrt{668627414781813055490\!-\!41706825763650297602\sqrt{257\,}\,}\,}\,}\,}\\[-10pt]& \ \!-\!4\sqrt{\begin{aligned}2570& \ \!+\!70\sqrt{257\,}\!+\!14\sqrt{514\!-\!2\sqrt{257\,}\,}\!+\!4\sqrt{57825\!-\!1985\sqrt{257\,}\!-\!\sqrt{2795782210\!-\!132283202\sqrt{257\,}\,}\,}\!+\!4\sqrt{181442\!+\!7022\sqrt{257\,}\!+\!2\sqrt{6216147922\!+\!319198766\sqrt{257\,}\,}\!+\!4\sqrt{4246573681\!+\!229781647\sqrt{257\,}\!+\!\sqrt{30446471517567866434\!+\!1879434198946564798\sqrt{257\,}\,}\,}\,}\\[-10pt]& \ \!-\!8\sqrt{\begin{aligned}122589& \ \!+\!5467\sqrt{257\,}\!+\!\sqrt{1610477650\!+\!75821486\sqrt{257\,}\,}\!+\!2\sqrt{2191066093\!+\!112238867\sqrt{257\,}\!+\!\sqrt{64430671830932914\!-\!4003622352651442\sqrt{257\,}\,}\,}\\[-10pt]& \ \!+\!2\sqrt{6266752250\!+\!360810454\sqrt{257\,}\!+\!2\sqrt{9280107224736158882\!+\!576107110307438686\sqrt{257\,}\,}\!+\!4\sqrt{4888235774502554705\!+\!303319711244318255\sqrt{257\,}\!+\!\sqrt{41347019429866775450521689114664709954\!+\!2579115892125395493567343388815241662\sqrt{257\,}\,}\,}\,}\end{aligned}\,}\end{aligned}\,}\end{aligned}\,} \end{split} } \style{text-align:right;}{ \begin{split} +\frac{1}{32}& \ \sqrt{\begin{aligned}\!-\!514& \ \!+\!2\sqrt{257\,}\!+\!2\sqrt{514\!-\!2\sqrt{257\,}\,}\!+\!4\sqrt{257\!+\!15\sqrt{257\,}\!+\!\sqrt{58082\!+\!3614\sqrt{257\,}\,}\,}\!+\!4\sqrt{514\!-\!18\sqrt{257\,}\!+\!6\sqrt{514\!-\!2\sqrt{257\,}\,}\!+\!4\sqrt{12593\!-\!561\sqrt{257\,}\!+\!\sqrt{21794114\!-\!552002\sqrt{257\,}\,}\,}\,}\\[-10pt]& \ \!-\!8\sqrt{257\!-\!\sqrt{257\,}\!-\!\sqrt{12850\!-\!754\sqrt{257\,}\,}\!+\!2\sqrt{3341\!-\!109\sqrt{257\,}\!-\!\sqrt{6509810\!-\!268274\sqrt{257\,}\,}\,}\!-\!2\sqrt{10794\!+\!70\sqrt{257\,}\!-\!2\sqrt{16703458\!-\!380386\sqrt{257\,}\,}\!-\!4\sqrt{4535793\!+\!253711\sqrt{257\,}\!-\!\sqrt{36803719442626\!+\!2282091196478\sqrt{257\,}\,}\,}\,}\,}\\[-10pt]& \ \!+\!8\sqrt{\begin{aligned}514& \ \!+\!14\sqrt{257\,}\!+\!6\sqrt{514\!-\!2\sqrt{257\,}\,}\!+\!4\sqrt{4369\!+\!15\sqrt{257\,}\!+\!\sqrt{9112706\!-\!354178\sqrt{257\,}\,}\,}\!-\!4\sqrt{4626\!+\!286\sqrt{257\,}\!+\!10\sqrt{247234\!+\!15422\sqrt{257\,}\,}\!+\!4\sqrt{3314529\!+\!206751\sqrt{257\,}\!+\!\sqrt{20768537086978\!+\!1295505749246\sqrt{257\,}\,}\,}\,}\\[-10pt]& \ \!+\!8\sqrt{3341\!+\!139\sqrt{257\,}\!+\!35\sqrt{514\!-\!2\sqrt{257\,}\,}\!+\!2\sqrt{1306845\!+\!53283\sqrt{257\,}\!-\!\sqrt{257924604274\!+\!4074182542\sqrt{257\,}\,}\,}\!-\!2\sqrt{3682810\!+\!172118\sqrt{257\,}\!+\!10\sqrt{12461428850\!+\!278588558\sqrt{257\,}\,}\!+\!4\sqrt{1185667505937\!+\!67520656239\sqrt{257\,}\!+\!\sqrt{375570632850308163054146\!+\!22763424315321766000318\sqrt{257\,}\,}\,}\,}\,}\end{aligned}\,}\\[-10pt]& \ \!+\!16\sqrt{\begin{aligned}257& \ \!-\!\sqrt{257\,}\!-\!\sqrt{514\!-\!2\sqrt{257\,}\,}\!-\!2\sqrt{257\!+\!15\sqrt{257\,}\!+\!\sqrt{58082\!+\!3614\sqrt{257\,}\,}\,}\!+\!6\sqrt{514\!-\!18\sqrt{257\,}\!+\!6\sqrt{514\!-\!2\sqrt{257\,}\,}\!+\!4\sqrt{12593\!-\!561\sqrt{257\,}\!+\!\sqrt{21794114\!-\!552002\sqrt{257\,}\,}\,}\,}\\[-10pt]& \ \!+\!4\sqrt{1285\!-\!5\sqrt{257\,}\!-\!5\sqrt{12850\!-\!754\sqrt{257\,}\,}\!+\!10\sqrt{3341\!-\!109\sqrt{257\,}\!-\!\sqrt{6509810\!-\!268274\sqrt{257\,}\,}\,}\!+\!2\sqrt{442554\!-\!13642\sqrt{257\,}\!+\!6\sqrt{4595699186\!-\!281900018\sqrt{257\,}\,}\!+\!4\sqrt{18581190721\!-\!1150035777\sqrt{257\,}\!+\!\sqrt{668627414781813055490\!-\!41706825763650297602\sqrt{257\,}\,}\,}\,}\,}\\[-10pt]& \ \!-\!4\sqrt{\begin{aligned}2570& \ \!+\!70\sqrt{257\,}\!+\!14\sqrt{514\!-\!2\sqrt{257\,}\,}\!+\!4\sqrt{57825\!-\!1985\sqrt{257\,}\!-\!\sqrt{2795782210\!-\!132283202\sqrt{257\,}\,}\,}\!+\!4\sqrt{181442\!+\!7022\sqrt{257\,}\!+\!2\sqrt{6216147922\!+\!319198766\sqrt{257\,}\,}\!+\!4\sqrt{4246573681\!+\!229781647\sqrt{257\,}\!+\!\sqrt{30446471517567866434\!+\!1879434198946564798\sqrt{257\,}\,}\,}\,}\\[-10pt]& \ \!+\!8\sqrt{\begin{aligned}122589& \ \!+\!5467\sqrt{257\,}\!+\!\sqrt{1610477650\!+\!75821486\sqrt{257\,}\,}\!+\!2\sqrt{2191066093\!+\!112238867\sqrt{257\,}\!+\!\sqrt{64430671830932914\!-\!4003622352651442\sqrt{257\,}\,}\,}\\[-10pt]& \ \!+\!2\sqrt{6266752250\!+\!360810454\sqrt{257\,}\!+\!2\sqrt{9280107224736158882\!+\!576107110307438686\sqrt{257\,}\,}\!+\!4\sqrt{4888235774502554705\!+\!303319711244318255\sqrt{257\,}\!+\!\sqrt{41347019429866775450521689114664709954\!+\!2579115892125395493567343388815241662\sqrt{257\,}\,}\,}\,}\end{aligned}\,}\end{aligned}\,}\end{aligned}\,}\end{aligned}\,} \end{split} }

References:
•  Lisl Gaal: Classical Galois Theory, 1973
•  Andreas Weber and Michael Keckeisen: Solving Cyclotomic Polynomials by Radical Expressions, 1999
•  Mariano Suárez-Álvarez: Algunas Construcciones, 2014 (personal communication)

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Adrian Turtschi, Jan 2008; updated Oct 2020