 In this video, we're going to prove that the classic geometric problem of doubling the cube is impossible. So what does it mean to double the cube? So given the edge of a cube, it is impossible to construct with a straight edge and a compass alone the edge of a cube that is twice the volume of the original cube. So when we think of a cube, we think of the following situation. So we draw our picture of a cube like so. And suppose that this cube has a side length of s and that this is a constructible number. The volume of this cube would then be s cubed. And so if we were to construct a cube whose volume is twice that. So the volume here is going to be twice that. So it's two times s cubed. We then want to draw this larger cube like so. What would be the side lengths of that cube? Okay, well the new cube, the duplicated cube, its volume needs to be two times s cubed. For which then, because volume is just whatever the side length is, we'll call it t for a moment. The volume is likewise t cubed. We can solve for t here very quickly and get we need the cube root of two times s cubed, which simplifies to be cube root of two times s. Now because the field of constructible numbers is itself a field, if we can construct s, we can divide by s. And therefore it requires we construct the cube root of two. So the side length of that cube is going to be the cube root of two here. Now in order to construct the cube root of two, we would need to be a constructible number, right? Now if you take q adjoin the cube root of two, this is a field here. If you look at the degree of the extension over q, this is going to be three because its minimum polynomial is x cubed minus two. But we proved previously that for the field of constructible numbers, if you take a constructible number and adjoin it to q, then the degree of the extension is always a power of two for which three is not a power of two. And that tells you that the cube root of two is not constructible and therefore we cannot double the cube.