Impossible Geometric Constructions

We show that it is impossible to use ruler and compasses to square the circle, double the cube, and trisect the angle. To prove this, we use concepts from modern algebra, specifically field theory.

00:00 Introduction
00:39 Ruler and compass constructions
04:57 The three problems and our strategy for showing their impossibility
12:36 Field extensions
13:33 Degree of field extension
17:35 Degrees of field extensions multiply
24:04 Field generated by alpha
26:22 Algebraic elements
29:15 Minimal polynomials
42:46 Irreducible monic polynomials are minimal polynomials
49:57 Degree of simple algebraic extensions
1:08:29 Constructible coordinates
1:27:47 Constructible coordinate a has deg(Q(a):Q) = 2^k
1:31:50 Squaring the circle is impossible
1:38:33 It is impossible to double every cube
1:47:02 It is impossible to trisect every angle

[lordlix]

That's was surprisingly easy to follow, your way of presenting the topics is very clear! I haven't had any actual abstract algebra class yet (but I know some definitions from self-learn), and I still managed to follow the entire video! This speaks itself of how good your presentation was! 😁

[Kavukamari]

what about squircling the cube three times?