An introduction to the modern theory of equations, by Florian Cajori.

82

THEORY OF EQUATIONS

Ex. 1. Find the primitive roots of x2 - 1 = 0 3-1 = 0,4 - 1 ==0.
Ex. 2. Find the roots ofxs - 1 = 0.
Dividing by x - 1, we get x4 + X3 + x2 + x + 1 = 0.
Dividing this reciprocal equation by x2 and taking x + - = y, we obtain
y2 + y = 1 and y =
Solving x2 - xy + 1 = 0, we arrive at the following four roots:
X1 =- I(1 + V5 + iV10 - 2v/5), x2 =- (1 - - /5 - iVi + 2V5),
Xa =-(- 5+ iV10 + 2/5), X4 =- -1(1 + V5-iV10 - 25).
These four are primitive fifth roots of unity. The other root is 1.
Show that x2 = xi2.
Ex. 3. Find the roots of x6 - 1 = 0.
Ex. 4. Find the roots of x7 - 1 = 0.
Dividing by x- 1, we get a reciprocal equation in the standard form
which can be depressed to the cubic y3 + y2 - 2 y - 1 = 0.
Writing z -= y + 1, we have Z3 - z-7 =0. By ~ 59 we obtain for
y three values, (, ai, a2, where
ac =- i + /28 + 84V- 3 + 1 " /28 - 84/v- 3.
From x2 - xy + 1 = 0 we get the six values
a( +x/ 2 - 4 Aa q /+ 2 -+ 4 ca2 VaC22- 4
2 '2 2
which, together with unity, are the seventh roots of unity.
Ex. 5. Find the roots of xs - 1. Which are primitive roots?
Ex. 6. Find the roots of x9 - 1 = 0.
Extracting the cube root, we get x3 =1 or w or wz2 and x = 1, w, 2,
/tvo, w /w,2, T/ow, T 0'w, w w'2, 2/w2, where w and w2 are the primitive cube roots of unity. Give the primitive roots of x9 - 1 = 0.
Ex. 7. Give a trigonometric solution of x10 - 1 = 0 and state which
roots are primitive.
Ex. 8. Find the primitive roots of x12 - 1 = 0.
Ex. 9. How many primitive roots has x180 - 1 = 0?
Ex. 10. Find the sum of the primitive roots of 14 - 1 = 0.