It is proper to distinguish the cases n prime and n composite; and in the latter case there is a distinction according as the prime factors of n are simple or multiple. By way of illustration, suppose successively n = 15 and n = 9; in the former case, if [alpha] be an imaginary root of x^3 - 1 = 0 (or root of x^2 + x + 1 = 0), and [beta] an imaginary root of x^5 - 1 = 0 (or root of x^4 + x^3 + x^2 + x + 1 = 0), then [omega] may be taken = [alpha][beta]; the successive powers thereof, [alpha][beta], [alpha]^2 [beta]^2, [beta]^3, [alpha][beta]^4, [alpha]^2, [beta], [alpha][beta]^2, [alpha]^2[beta]^3, [beta]^4, [alpha], [alpha]^2 [beta], [beta]^2, [alpha][beta]^3, [alpha]^2 [beta]^4, are the roots of x^14 + x^13 + ... + x + 1 = 0; the solution thus depends on the solution of the equations x^3 - 1 = 0 and x^5 - 1 = 0. In the latter case, if [alpha] be an imaginary root of x^3 - 1 = 0 (or root of x^2 + x + 1 = 0), then the equation x^9 - 1 = 0 gives x^3 = 1, [alpha], or [alpha]^2; x^3 = 1 gives x = 1, [alpha], or [alpha]^2; and the solution thus depends on the solution of the equations x^3 - 1 = 0, x^3 - [alpha] = 0, x^3 - [alpha]^2 = 0. The first equation has the roots 1, [alpha], [alpha]^2; if [beta] be a root of either of the others, say if [beta]^3 = [alpha], then assuming [omega] = [beta], the successive powers are [beta], [beta]^2, [alpha], [alpha][beta], [alpha][beta]^2, [alpha]^2, [alpha]^2[beta], [alpha]^2 [beta]^2, which are the roots of the equation x^8 + x^7 + ... + x + 1 = 0. It thus appears that the only case which need be considered is that of n a prime number, and writing (as is more usual) r in place of [omega], we have r, r^2, r^3, ... r^(n - 1) as the (n - 1) roots of the reduced equation x^(n - 1) + x^(n - 2) + ... + x + 1 = 0; then not only r^n - 1 = 0, but also r^(n - 1) + r^(n - 2) + ... + r + 1 = 0. 23. The process of solution due to Karl Friedrich Gauss (1801) depends essentially on the arrangement of the roots in a certain order, viz. not as above, with the indices of r in arithmetical progression, but with their indices in geometrical progression; the prime number n has a certain number of prime roots g, which are such that g^(n - 1) is the lowest power of g, which is [equivalent to] 1 to the modulus n; or, what is the same thing, that the series of powers 1, g, g^2, ... g^(n - 2), each divided by n, leave (in a different order) the remainders 1, 2, 3, ... n - 1; hence giving to r in succession the indices 1, g, g^2, ... g^(n - 2), we have, in a different order, the whole series of roots r, r^2, r^3, ... r^(n - 1).