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"Using straight edge and compass you can only solve equations with x to the power of 2." Why is that?

• hypersoar 12y ago
  First, what does "solving" an equation mean in this context? Start with a pair of points, defined to be distance one apart. To solve an equation for x is to construct a line segment of length x (i.e., a line segment of length x multiplied by the length of that "unit" segment). For example, one can solve the equation x^2 = 2 by constructing a unit square and then connecting the opposite corners. With a compass and straightedge, you can make two things: straight lines and circles, which are defined by linear and quadratic polynomials, respectively. Any constructible length will then be a solution to a bunch of equations made from various linear and quadratic polynomials. And these are the only lengths you can make. You can't make a cubic polynomial by plugging linear and quadratic polynomials into each other (you can make it by multiplying polynomials, but that's not what we're doing, here). The reason "trisecting the angle" is impossible is because it involves solving a cubic polynomial. One can, however, solve polynomials whose degree is a power of two. The way to do this rigorously is with field extensions and Galois theory.