The classical Greek questions
October 18, 2011 4 Comments
Today we’ll answer these questions:
- Can we build a cube with twice the volume of a given cube?
- Can we trisect an arbitrary angle?
- Can we construct a square with the same area as the unit circle?
Remember that . We’re considering as a vector space over the field , and want to know its dimension. Let be an irreducible polynomial with and . The vector space is definitely spanned by . I claim that form a basis. This would be a basis with elements, making .
We have a linear dependence , so any basis must have fewer than elements. Now suppose we have a linear dependence among . Say,
Then we have a polynomial of smaller degree with .
Why is this a problem? We can take combinations of and , and try to make the polynomial of smallest degree in this way. Let’s call it for some . This is the “greatest common divisor” of and . You thought GCDs only existed for integers? Nope. There’s a general structure (of which the integers are an example) which always has a concept of a GCD.
The important thing is that is a divisor of . But we assumed was irreducible, so the only possible divisors are itself and . Since has smaller degree than , so does , so it must be that . But then . What? Contradiction!
This means we have is a basis, so .
This “back-and-forth” between bases for vector spaces and polynomials is a really cool trick, and is a fun part of Galois theory.
Now we can answer the questions.
- NO! If we could construct a cube with twice the volume of a given cube, we could in particular construct a cube of volume 2, so we could construct the side length . But is an irreducible polynomial (check that yourself) with as a root, so this would mean we have a degree 3 extension . But every extension of in has degree that is a power of two. Three is certainly not a power of two. Contradiction!
- NO! If we could construct arbitrary angles given , we could in general construct the length from . If , then , which is a root of the polynomial (this comes from the triple-angle formula for cosine).
- NO! If we could, we could construct the side length . This is the most absurd, because then we could construct which isn’t the root of any polynomial. Some might want to even write , which is accurate. We can’t get to infinity by only finite constructions.