Why are Most Polygons Impossible to Construct?

Video Description

⬣ LINKS ⬣ ⬡ PATREON: https://www.patreon.com/anotherroof ⬡ CHANNEL: https://www.youtube.com/c/AnotherRoof ⬡ WEBSITE: https://anotherroof.top ⬡ SUBREDDIT: https://www.reddit.com/r/anotherroof ⬡ TWITCH: https://www.twitch.tv/anotherroof ⬣ 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