 So to construct the heptadecagon, we need to consider the roots of z to the 17th minus 1 equals 0. Gauss's approach would be to break the roots into sets whose sums, f lambda, are solutions to an equation, solve the equation, then lather rins repeat for each f lambda. But will this always work? Well, let's consider the heptagon. Let's try to find the seventh root of unity. So if r is the seventh root of unity, then one partition of the roots will be, and again, let's make these the roots of an equation. Our equation will be, and if we expand that out, and we can simplify, we know it 2-1, 2-2, and 2-3 are. So this first coefficient is just going to be the sum of all those powers. For the next coefficient, we'll have to find a few products. 2-1 times 2-2 is 2-1 times 2-3, 2-2 times 2-3, and the product 2-1, 2-2, 2-3 will be, and again, remember that the sum of all the powers is negative 1, and so we can isolate those sets that are sums of all the powers. Also remember that these are the seventh roots of unity, so the bracket 7 is just equal to 1. And since this is a cubic equation, we can solve it. Or can we? And there's just one problem. We're talking about straight-edge and compass constructability, and straight-edge and compass constructions can be used to locate the intersection of two lines, a line in a circle, or two circles. But all three cases reduce to solving linear or quadratic equations. And you might remember that to solve a cubic, we need to introduce conic sections, which are not part of compass and straight-edge constructability. Well, maybe we just found the wrong period. What if we find a different period? If r is a seventh root of unity, a different partition of the roots will be, and again, this time we'll let 3-1 and 3-3 be the solutions, and simplify, and since this is a quadratic, its solutions correspond to something we can do with compass and straight-edge. Unfortunately, this doesn't help us because there's a next step. In order to isolate one of the roots, we'll suppose one of our solutions is, say, 1 plus 2 plus 4, the individual roots will be the solution to the equation, which will be a cubic equation and unsolvable using compass and straight-edge. So now let's think about it. If n is prime, there are n minus 1 primitive roots, and Gauss's procedure allows us to split these into sets with equal numbers of roots. Now for n equals 7, the 7 minus 1 6 primitive roots split into either 3 sets of 2, which were solutions to a cubic equation, and each set of 2 was a solution to a quadratic equation. And so we could construct the heptagon by solving a cubic equation and a quadratic equation, which means we'll have to use conic sections. And you might remember the Islamic Geometers did exactly that. Or we could also split the roots into 2 sets of 3, which were solutions to a quadratic equation, and each element of the set was the solution to a cubic equation. And in other words, a cubic equation is unavoidable. On the other hand, if n equals 5, the 5 minus 1 4 primitive roots split into 2 sets of 2, which were solutions to a quadratic equation, and each element of the set of 2 was a solution to another quadratic equation. And so we could construct the pentagon by solving two quadratic equations. And because these are quadratic, this can be done using compass and straightedge. So what is it that 7 minus 1 6 has that 5 minus 1 4 doesn't? And the answer is that 6 has a prime factor other than 2. So remember Fermat conjectured wrongly that 2 to the 2 to the k plus 1 was always prime. It sometimes is. And so the 2 to the 2 to the 2 plus 1 17 roots of unity yield 17 minus 1 16 primitive roots. And Gauss's method allows us to split these into 2 sets of 8, which were solutions to a quadratic equation. Then each set of 8 could be split into 2 sets of 4, which were solutions to a quadratic equation. And each set of 4 could be split into 2 sets of 2, which were solutions to a quadratic equation. And each element of the set of 2 was a solution to a quadratic equation. And so we could find the primitive 17th root by solving several quadratic equations and construct the heptidecagon by compass and straightedge. Now we usually say that Gauss found a method of constructing the regular heptidecagon, but in fact he showed that it was possible but didn't produce the actual construction. And a number of constructions did emerge during the course of the 19th century.