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3.5. Roots of Unity 101

3.5 Roots of Unity

The factorization of t” - 1 was studied by C.F. Gauss (1777-1855)) who
wanted to find out the condition on n in order that a regular n-gon could
be constructed using ruler and compasses.
From de Moivre’s Theorem, it is possible to locate all the zeros of this
polynomial as equally spaced points on the circumference of the unit circle
in the complex plane (i.e. as vertices of a regular n-gon). From this it is a
simple step to derive the decomposition oft” - 1 into linear factors over C.
These factors can then be combined to yield factorizations over R and Z.
In this section, we will denote tn - 1 by P,(t).

Exercises

1. Roots of unity. Let n be a positive integer, and let r(cos B + i sin 0) be
   a root of the polynomial P,(t). Use de Moivre’s Theorem (Exercise
   1.3.8) to show that

r”(cosn0 + isinn0) = 1.

Take absolute values of both sides and deduce that r = 1. Show that

n0 must be an integer multiple of 27r.

Show that a complete set of zeros of P,(t) consists of the complex

numbers,

1, cos 2x/n + i sin 27r/n, cos 47rln + i sin 4x/n,... ,

cos

2(n - 1)7r + isin 2(n - ‘>~

n n ’

These are called the nth roots of unity.

Draw in the complex plane the unit circle (center 0 and radius 1))

and indicate on this circle the location of the nth roots of unity.

2. Show that the factorization of P,,(t) over C into a product of irre-
   ducible polynomials is given by


3. Factor over C the following polynomials: P2, P3, Pa, Ps, Pa. Express
   all coefficients in the form a + bi where a, b are real.