Why are Most Polygons Impossible to Construct?

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⬣ ABOUT ⬣
Only certain regular polygons are constructible with compass and straightedge. Why? And why did the first person to prove it after 2000 years get no recognition? In this introduction to Field Theory, we’ll find out, and along the way we’ll also prove other impossibilities like why cube doubling and angle trisection are impossible. Happy Birthday Pierre Wantzel!

⬣ TIMESTAMPS ⬣
00:00 - Intro
02:13 - Why ONLY the four operations and Square Roots
04:14 - Field Extensions
14:37 - Minimal Polynomials
25:54 - Cube Doubling is Impossible
31:50 - Angle Trisection is Impossible
36:35 - Constructible Polygons: The Gauss-Wantzel Theorem
50:12 - Why Pierre Wanzel got No Recognition

⬣ INVESTIGATORS ⬣
Nothing for you here. Sorry!

⬣ REFERENCES ⬣
Biographical details on Wantzel to be found in:
Jean-Claude Saint-Venant (1848). “Biographie: Wantzel”. Nouvelles Annales de Mathématiques Série 1, 7: 321-331.
Found, with English translation by Lauren Murphy, here: https://divisbyzero.com/wp-content/uploads/2018/02/saint-venant.pdf

Amazing treatise on possible reasons why Wantzel was never credited:
Jesper Lützen, “Why was Wantzel Overlooked for a Century? The Changing Importance of an Impossibility Result” Historia Mathematica, Vol 36 (4) 374-394, 2009.
Found here: https://www.sciencedirect.com/science/article/pii/S031508600900010X

Pierre Wantzel’s original paper, with proofs adapted in the video:
Wantzel, L. (1837), "Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas", Journal de Mathématiques Pures et Appliquées, 2: 366–372.
Found here: http://visualiseur.bnf.fr/ConsulterElementNum?O=NUMM-16381&Deb=374&Fin=380&E=PDF

⬣ CREDITS ⬣
Music by Danijel Zambo, Tobias Voigt, and Apex Music.

Image Credits
Wantzel
https://mathshistory.st-andrews.ac.uk/Biographies/Wantzel/poster/lived/
Hamilton
https://upload.wikimedia.org/wikipedia/commons/f/f6/William_Rowan_Hamilton_portrait_oval_combined.png
Klein
https://upload.wikimedia.org/wikipedia/commons/4/4e/Felix_Klein%2C_ante_1897_-_Accademia_delle_Scienze_di_Torino_0078_B.jpg
De Morgan
https://upload.wikimedia.org/wikipedia/commons/2/2c/De_Morgan_Augustus.jpg
Petersen
https://upload.wikimedia.org/wikipedia/commons/6/65/Petersen_Julius.jpg
Pascal's Triangle
https://www.geogebra.org/resource/jk42jcvy/wp6YMh81jzMYSh2D/material-jk42jcvy.png