On the solution of the equation xp–1=0 (p being a prime number)

Abstract

[At the first meeting of this Session a paper was read on the value of cos 2π/17, which evidently may be made to depend on the solution of x17 – 1 = 0.* The present paper is the outcome of a suggestion then made, that a sketch of Gauss's treatment of the general equation might prove interesting. To give completeness to the subject the necessary theorems on congruences have been prefixed. The convenient notation introduced by Gauss is here adopted; thus, when the difference between a and b is divisible by p, instead of writing a = Mp + b, we may write a ≡ b (mod p), the value of M seldom being of importance. It is evident that if a ≡ b, then na ≡ nb, and an ≡ bn, n being any positive integer, and the same modulus p being understood throughout. Also a/n ≡ b/n provided n be prime to p. Other properties (similar to those of equations) are easily seen, but only the above are needed here.