Cyclotomic polynomial on arvau

Wikipedia / Cyclotomic polynomial

In mathematics-th cyclotomic polynomial, for any positive integer, is the unique irreducible polynomial with integer coefficients that is a divisor. Its roots-th primitive roots of unity and coprime is the imaginary unit-th cyclotomic polynomial is equal to

It may also be defined as the monic polynomial with integer coefficients that is the minimal polynomial over the field of the rational numbers of any primitive nth-root of unity is an example of such a root).

An important relation linking cyclotomic polynomials and primitive roots of unity is

Examples

If n is a prime number, then

If n = 2p where p is a prime number other than 2, then

For n up to 30, the cyclotomic polynomials are:

The case of the 105th cyclotomic polynomial is interesting because 105 is the least positive integer that is the product of three distinct odd prime numbers (3×5×7) and this polynomial is the first one that has a coefficient other than 1, 0, or −1:

Properties

Fundamental tools

The cyclotomic polynomials are monic polynomials with integer coefficients that are irreducible over the field of the rational numbers. Except for n equal to 1 or 2, they are palindromess of even degree.

, or in other words the number of n is Euler's totient function.

in the ring is a nontrivial result due to Gauss. Depending on the chosen definition, it is either the value of the degree or the irreducibility which is a nontrivial result. The case of prime n is easier to prove than the general case, thanks to Eisenstein's criterion.

A fundamental relation involving cyclotomic polynomials is

which means that each n-th root of unity is a primitive d-th root of unity for a unique d dividing n.

The Möbius inversion formula to be expressed as an explicit rational fraction:

is the Möbius function.

This provides a recursive formula, which may be computed by dividing for the proper divisors d dividing n:

, provided integer factorization and division of polynomials are available. Many computer algebra systems, such as SageMath, Maple, Mathematica, and PARI/GP, have a built-in function to compute the cyclotomic polynomials.

Easy cases for computation

As noted above, if is a prime number, then

If n is an odd integer greater than one, then

In particular, if is twice an odd prime, then (as noted above)

If is a prime power (where p is prime), then

More generally, if with relatively prime to , then

in terms of a cyclotomic polynomial of square free index: If is the product of the prime divisors of (its radical), then

This allows formulas to be given for the th cyclotomic polynomial when has at most one odd prime factor: If is an odd prime number, and and are positive integers, then

For other values of , the computation of the where is the product of the distinct odd prime divisors of . To deal with this case, one has that, for prime and not dividing ,

Integers appearing as coefficients

The problem of bounding the magnitude of the coefficients of the cyclotomic polynomials has been the object of a number of research papers.

If n are all in the set {1, −1, 0}.

it has a coefficient −2 (see above only has coefficients in {1, −1, 0}.

If n, the smallest n with 6 different odd primes, has coefficients of magnitude up to 532.

Let A(n) denote the maximum absolute value. It is known that for any positive k, the number of n up to x with A(n) > nk is at least c(k)⋅x for a positive c(k) depending on k and x sufficiently large. In the opposite direction, for any function ψ(n) tending to infinity with n we have A(n) bounded above by nψ(n) for almost all n.

A combination of theorems of Bateman and Vaughan states that, we have

, and on the other hand, we have

. This implies in particular that univariate polynomials) whose coefficients are superpolynomially larger than the original coefficients. This is not too far from the general Landau-Mignotte bound.

Gauss's formula

Let n be odd, square-free, and greater than 3. Then:

for certain polynomials An(z) and Bn(z) with integer coefficients, An(z) of degree (n)/2, and Bn(z) of degree (n)/2 − 2. Furthermore, An(z) is palindromic when its degree is even; if its degree is odd it is antipalindromic. Similarly, Bn(z) is palindromic unless n is composite and n ≡ 3 (mod 4), in which case it is antipalindromic.

The first few cases are

Lucas's formula

Let n be odd, square-free and greater than 3. Then

for certain polynomials Un(z) and Vn(z) with integer coefficients, Un(z) of degree (n)/2, and Vn(z) of degree (n)/2 − 1. This can also be written

If n is even, square-free and greater than 2 (this forces n/2 to be odd),

for Cn(z) and Dn(z) with integer coefficients, Cn(z) of degree (n), and Dn(z) of degree (n) − 1. Cn(z) and Dn(z) are both palindromic.

The first few cases are:

Sister Beiter conjecture

The Sister Beiter conjecture of coefficients of ternary cyclotomic polynomials are three odd primes.

Cyclotomic polynomials over a finite field and over the p-adic integers

Over a finite field with a prime number of elements, for any integer that is not a multiple of irreducible polynomials of degree is Euler's totient function and is the multiplicative order of modulo is irreducible if and only if is a primitive root modulo n, that is, does not divide , and its multiplicative order modulo .

These results are also true over the p-adic integers, since Hensel's lemma allows lifting a factorization over the field with elements to a factorization over the -adic integers.

Polynomial values

If for every (this follows from the fact that the roots of a cyclotomic polynomial are all non-real, for ).

For studying the values that a cyclotomic polynomial may take when is given an integer value, it suffices to consider only the case , as the cases and ).

For , one has

if is not a prime power,
is a prime power with .

may take for other integer values of is strongly related with the multiplicative order modulo a prime number.

More precisely, given a prime number and an integer coprime with , the multiplicative order of modulo , is the smallest positive integer such that For , the multiplicative order of modulo is also the shortest period of the representation of in the numeral base (see Unique prime; this explains the notation choice).

The definition of the multiplicative order implies that, if is the multiplicative order of modulo , then The converse is not true, but one has the following.

If is a positive integer and is an integer, then (see below for a proof)

where

is a non-negative integer, always equal to 0 when is even. (In fact, if is neither 1 nor 2, then is either 0 or 1. Besides, if is not a power of 2, then is always equal to 0)
is 1 or the largest odd prime factor of .
is odd, coprime with , and its prime factors are exactly the odd primes such that is the multiplicative order of modulo .

This implies that, if then either is a divisor of or is a divisor of

Zsigmondy's theorem implies that the only cases where and are

It follows from above factorization that the odd prime factors of

are exactly the odd primes such that is the multiplicative order of modulo . This fraction may be even only when is odd. In this case, the multiplicative order of modulo is always .

There are many pairs with is prime. In fact, Bunyakovsky conjecture implies that, for every , there are infinitely many is prime. See for the list of the smallest is prime (the smallest is Euler–Mascheroni constant is Euler's totient function). See also with and , and, more generally, , for the smallest positive integers of this form.

Values of is a prime power, then
If and for dividing and different of . If is a prime divisor of multiplicity in divide , and their values at are factors equal to As is the multiplicity of in , cannot divide the value at
If is the multiplicative order of modulo , then then showing that, if there would be the case, would not be the multiplicative order of modulo .
The other prime divisors of are divisors of . Let such that is not be the multiplicative order of modulo . If is the multiplicative order of modulo , then The resultant where and are polynomials. Thus divides this resultant. As divides , and the resultant of two polynomials divides the discriminant of any common multiple of these polynomials, Thus divides .
and are coprime. In other words, if is a prime common divisor of then is not the multiplicative order of modulo . By Fermat's little theorem, the multiplicative order of is a divisor of , and thus smaller than .
is square-free. In other words, if is a prime common divisor of Let does not divide for some polynomial We take
The multiplicative order of modulo divides , which is a divisor of . Thus is a multiple of . Now,
As

Applications

, one can give an elementary proof for the infinitude of primess congruent to 1 modulo n, which is a special case of Dirichlet's theorem on arithmetic progressions.

. Since

we have

is a double root of

must be a root of the derivative so

Periodic recursive sequences

The constant-coefficient linear recurrences which are periodic are precisely the power series coefficients of rational functions whose denominators are products of cyclotomic polynomials.

In the theory of combinatorial generating functions, the denominator of a rational function determines a linear recurrence for its power series coefficients. For example, the Fibonacci sequence has generating function and equating coefficients. has a recursive sequence of coefficients which is periodic with period at most n. For example, , so we may write , and the sequence has period 6 with initial values given by the coefficients of the numerator.