| Gauss's theorem states that a Euclidean construction can be done if and only if the number of sides of a regular n-sided polygon may be expressed as the product of a power of two and distinct Fermat primes. That is a Euclidean construction is possible if n = 2ip0p2...p j. where n ≥ 3, i ≥ 0, j ≥ 0, and p0, p2, ..., pj are distinct Fermat primes. This paper will utilize Galois Theory to prove of the necessity section of the theorem. The sufficiency of the proof will completed using properties of number theory relating to integer solutions of Xn - 1 = 0.; This paper will begin with the necessary background for developing Galois theory, then proceed the to the required principles of Galois theory and number theory for providing a proof for the Gauss Theorem. The proof of Gauss's Theorem will be given and then an open question related to the proof will be presented. |
