Roots of Unity: The Cyclotomic Polynomial

Since 1 is always a root of unity, we can determine the other roots of unity by dividing (xⁿ - 1) by (x -1) and solving the equation.

In the case of n=2, this gives us: (x² - 1)/(x - 1) = (x + 1). So, the other root of unity is x = -1.

In the case of n=3, this gives us: (x³ - 1)/(x - 1) = (x² + x + 1). Using the quadratic equation to solve for this (see Theorem, here) gives us:

(-1 ± √1 - 4)/2 = (-1 ± √-3)/2.

In light of this and a previous result (see Lemma 1, here), it is possible to generate an equation to determine the primitive nth roots of unity.

Definition 1: Cyclotomic Polynomial Φ_n(x)

Φ₁(x) = x - 1

Otherwise:



where d includes all divisors of n except for n itself.

Lemma 1:

If p is prime, then:

Φ_p = x^p-1 + x^p-2 + ... + x + 1

Proof:

(1) Since p is prime, the only divisor is 1 and:

Φ_p = (x^p - 1)/(x - 1)

(2) Now, we know that (x^p - 1)/(x - 1) = x^p-1 + x^p-2 + ... + x + 1 since:

(a) Let s = x^p-1 + x^p-2 + ... + x + 1

(b) x*s = x^p + x^p-1 + ... + x² + x

(c) x*s - s = x^p - 1

(d) s(x - 1) = x^p - 1

(e) s = (x^p - 1)/(x - 1)

QED

Now, the next thing we need is a proof that Φ_n will generate a polynomial whose solution is the primitive n-th roots of unity:

Theorem 2: Cyclotomic Polynomial

For every integer n ≥ 1, Φ_n is a monic polynomial (see Definition 5, here for monic if needed) with integral coefficients and:

Φ_n = ∏(x - ζ) ∈ Z[x]

where ζ runs over the set of primitive n-th roots of unity and Z is the set of integers.

Proof:

(1) For Φ₁=x-1, the theorem is clearly true.

(2) Let's assume that the theorem is true up to n-1.

(3) Since d is always less than n, for each d, we have:

Φ_d(x) = ∏(x - ζ) ∈ Z[x]

where ζ runs over the roots of unity with exponent d.

(4) Since each root of Φ_d is a root of unity, it follows that Φ_d divides xⁿ - 1 [see Lemma 1, here]

(5) If d₁, d₂ are distinct divisors of n, then it follows that Φ_d1 and Φ_d2 are relatively prime polynomials and distinct -- that is, they do not include any common root of unity since:

Each root of unity has a unique exponent so no root of unity included in Φ_n will be found in more than one divisor Φ_di.

(6) Using Corollary 2.2, here, it follows that:



is a polynomial since it is divible by a product of all Φ_d

(7) Since each Φ_d is monic, it follows that Φ_n is monic.

(8) Since each Φ_d is in Z[x] and since xⁿ - 1 is in Z[x], using the Division Algorithm for Polynomials, it follows that Φ_n is in Z[x]. [See Theorem, here]

(9) Now for each (x - ζ_i) that divides Φ_n, it follows that it's exponent equals n since:

(a) If its exponent is less than n and it is an n-th root of unity, then its exponent e is a divisor of n and it has been divided out.

(b) Assume that e doesn't divide n.

(c) So that n = qe + r where r ≥ 1 and r is less than e.

(d) Then:

(ζ)^qe+r = 1 = (ζ)^qe*(ζ^r)

(e) We know that (ζ)^qe = 1 since:

(ζ)^qe = (ζ^e)^q

(f) But then:

ζ^r = 1/(ζ^qe) = 1

(g) This is impossible since r is less than e and e is the exponent of the root of unity.

(h) Therefore, we reject the assumption in 9b.

QED

References

• Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, 2001