22. It has already been shown how the several roots of the equation xn − 1 = 0 can be expressed in the form cos 2sπ/n + i sin 2sπ/n, but the question is now that of the algebraical solution (or solution by radicals) of this equation. There is always a root = 1; if ω be any other root, then obviously ω, ω², ... ωn−1 are all of them roots; xn − 1 contains the factor x − 1, and it thus appears that ω, ω², ... ωn−1 are the n-1 roots of the equation
xn−1 + xn−2 + ... x + 1 = 0;
we have, of course, ωn−1 + ωn−2 + ... + ω + 1 = 0.
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 α be an imaginary root of x³ − 1 = 0 (or root of x² + x + 1 = 0), and β an imaginary root of x5 − 1 = 0 (or root of x4 + x³ + x² + x + 1 = 0), then ω may be taken = αβ; the successive powers thereof, αβ, α²β², β³, αβ4, α², β, αβ², α²β³, β4, α, α²β, β², αβ³, α²β4, are the roots of x14 + x13 + ... + x + 1 = 0; the solution thus depends on the solution of the equations x³ − 1 = 0 and x5 − 1 = 0. In the latter case, if α be an imaginary root of x³ − 1 = 0 (or root of x² + x + 1 = 0), then the equation x9 − 1 = 0 gives x³ = 1, α, or α²; x³ = 1 gives x = 1, α, or α²; and the solution thus depends on the solution of the equations x³ − 1 = 0, x³ − α = 0, x³ − α² = 0. The first equation has the roots 1, α, α²; if β be a root of either of the others, say if β³ = α, then assuming ω = β, the successive powers are β, β², α, αβ, αβ², α², α²β, α²β², which are the roots of the equation x8 + x7 + ... + 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 ω, we have r, r², r³,...rn−1 as the (n − 1) roots of the reduced equation
xn−1 + xn−2 + ... + x + 1 = 0;
then not only rn − 1 = 0, but also rn−1 + rn−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 gn−1 is the lowest power of g, which is ≡ 1 to the modulus n; or, what is the same thing, that the series of powers 1, g, g², ... gn−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, g²,...gn−2, we have, in a different order, the whole series of roots r, r², r³,...rn−1.
In the most simple case, n = 5, the equation to be solved is x4 + x³ + x² + x + 1 = 0; here 2 is a prime root of 5, and the order of the roots is r, r², r4, r³. The Gaussian process consists in forming an equation for determining the periods P1, P2, = r + r4 and r² + r³ respectively;—these being such that the symmetrical functions P1 + P2, P1P2 are rationally determinable: in fact P1 + P2 = −1, P1P2 = (r + r4) (r² + r³), = r³ + r4 + r6 + r7, = r³ + r4 + r + r², = −1. P1, P2 are thus the roots of u² + u − 1 = 0; and taking them to be known, they are themselves broken up into subperiods, in the present case single terms, r and r4 for P1, r² and r³ for P2; the symmetrical functions of these are then rationally determined in terms of P1 and P2; thus r + r4 = P1, r·r4 = 1, or r, r4 are the roots of u² − P1u + 1 = 0. The mode of division is more clearly seen for a larger value of n; thus, for n = 7 a prime root is = 3, and the arrangement of the roots is r, r³, r², r6, r4, r5. We may form either 3 periods each of 2 terms, P1, P2, P3 = r + r6, r³ + r4, r² + r5 respectively; or else 2 periods each of 3 terms, P1, P2 = r + r² + r4, r³ + r6 + r5 respectively; in each ease the symmetrical functions of the periods are rationally determinable: thus in the case of the two periods P1 + P2 = −1, P1P2 = 3 + r + r² + r³ + r4 + r5 + r6, = 2; and the periods being known the symmetrical functions of the several terms of each period are rationally determined in terms of the periods, thus r + r² + r4 = P1, r·r² + r·r4 + r²·r4 = P2, r·r²·r4 = 1.
The theory was further developed by Lagrange (1808), who, applying his general process to the equation in question, xn−1 + xn−2 + ... + x + 1 = 0 (the roots a, b, c... being the several powers of r, the indices in geometrical progression as above), showed that the function (a + ωb + ω²c + ...)n−1 was in this case a given function of ω with integer coefficients.