Cyclotomic polynomial

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In algebra, the n^th cyclotomic polynomial, for any positive integer n, is the monic polynomial

Φn(X) =	∏	(X − ω)
	ω

where the product is over all primitive n^th roots of unity ω, i.e. all the complex numbers ω of order n.

Contents 1 Properties 2 Examples 3 Applications 4 See also 5 References

[edit] Properties

The degree of Φ_n, or in other words the number of factors in its definition above, is φ(n), where φ is Euler's totient function.

The coefficients of Φ_n are integers. In other words,

For any positive integer n we have

and (what is equivalent, by Möbius inversion)

where μ is the Möbius function.

The polynomial Φ_n(X) is irreducible in the ring . This result, due to Gauss, is not trivial.^[1] The case of prime n is easier to prove than the general case, thanks to Eisenstein's criterion.

If n is a prime power, say p^m where p is prime, then

In particular ( for m = 1)

Any cyclotomic polynomial Φ_n(X) has a simple expression in terms of Φ_q(X) where q is the radical of n:

Φ_n(X) = Φ_q(X^{n / q})

If n > 1 is odd, then Φ_2n(X) = Φ_n( − X).

If n has at most two distinct odd prime factors, then the coefficients of Φ_n are all in the set {1, −1, 0}. The converse is not true. For instance, only has coefficients in {1, −1, 0}. The first cyclotomic polynomial having a coefficient not in this set is Φ₁₀₅(X) with coefficient −2.

[edit] Examples

Φ₁(X) = X − 1

Φ₂(X) = X + 1

Φ₃(X) = X² + X + 1

Φ₆(X) = X² − X + 1

Φ₉(X) = X⁶ + X³ + 1

Φ₁₅(X) = X⁸ − X⁷ + X⁵ − X⁴ + X³ − X + 1

[edit] Applications

Using the fact that Φ_n is irreducible, one can prove the existence of a prime congruent to 1 modulo n,^{[citation needed]} which is a special case of Dirichlet's theorem on arithmetic progressions.

[edit] See also

• Cyclotomic field
• Root of unity

[edit] References

1. ^ Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, MR1878556, ISBN 978-0-387-95385-4