Cyclotomic polynomial

Cyclotomic polynomial
In mathematics, the nth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients that is a divisor of and is not a divisor of for any Its roots are all nth primitive roots of unity
, where k runs over the positive integers not greater than n and coprime to n (and i is the imaginary unit). In other words, the nth 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
showing that is a root of if and only if it is a dth primitive root of unity for some d that divides n.
Examples
If n is a prime number, then
If n = 2p where p is an odd prime number, 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 palindromes of even degree.
The degree of , or in other words the number of nth primitive roots of unity, is , where is Euler's totient function.
The fact that is an irreducible polynomial of degree 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 allows the expression of as an explicit rational fraction:
where is the Möbius function.
The cyclotomic polynomial may be computed by (exactly) dividing by the cyclotomic polynomials of the proper divisors of n previously computed recursively by the same method:
(Recall that .)
This formula defines an algorithm for computing for any 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
These formulas may be applied repeatedly to get a simple expression for any cyclotomic polynomial in term of a cyclotomic polynomial of square free index: If is the product of the prime divisors of (its radical), then
This allows to give formulas 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 the other values of , the computation of the th cyclotomic polynomial is similarly reduced to that of 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. Several survey papers give an overview.
If n has at most two distinct odd prime factors, then Migotti showed that the coefficients of are all in the set {1, −1, 0}.
The first cyclotomic polynomial for a product of three different odd prime factors is it has a coefficient −2 (see its expression above). The converse is not true: only has coefficients in {1, −1, 0}.
If n is a product of more different odd prime factors, the coefficients may increase to very high values. E.g., has coefficients running from −22 to 23 the smallest n with 6 different odd primes, has coefficients of magnitude up to 532.
Let A(n) denote the maximum absolute value of the coefficients of Φn. 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 resp. Vaughan states that on the one hand, for every , we have
for all sufficiently large positive integers , and on the other hand, we have
for infinitely many positive integers . This implies in particular that univariate polynomials (concretely for infinitely many positive integers ) can have factors (like ) 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:
where both An(z) and Bn(z) have integer coefficients, An(z) has degree φ(n)/2, and Bn(z) has degree φ(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 ≡ 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
where both Un(z) and Vn(z) have integer coefficients, Un(z) has degree φ(n)/2, and Vn(z) has degree φ(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),
where both Cn(z) and Dn(z) have integer coefficients, Cn(z) has degree φ(n), and Dn(z) has degree φ(n) − 1. Cn(z) and Dn(z) are both palindromic.
The first few cases are:
Sister Beiter conjecture
The Sister Beiter conjecture is concerned with the maximal size (in absolute value) of coefficients of ternary cyclotomic polynomials where are three prime numbers.
Cyclotomic polynomials over a finite field and over the -adic integers
Over a finite field with a prime number of elements, for any integer that is not a multiple of , the cyclotomic polynomial factorizes into irreducible polynomials of degree , where is Euler's totient function and is the multiplicative order of modulo . In particular, is irreducible if and only if is a primitive root modulo , that is, does not divide , and its multiplicative order modulo is , the degree of .
These results are also true over the -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 takes any real value, then 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 are trivial (one has and ).
For , one has
if is not a prime power,
if is a prime power with .
The values that a cyclotomic polynomial 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 is a divisor of 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 is a divisor of 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 is an odd prime divisor of then either is a divisor of or is a divisor of . In the latter case, does not divide
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 such that is prime. In fact, Bunyakovsky conjecture implies that, for every , there are infinitely many such that is prime. See for the list of the smallest such that is prime (the smallest such that is prime is about , where is Euler–Mascheroni constant, and is Euler's totient function). See also for the list of the smallest primes of the form with and , and, more generally for the smallest positive integers of this form.
Values of If is a prime power, then
If is not a prime power, let we have and is the product of the for dividing and different of . If is a prime divisor of multiplicity in , then divide , and their values at are factors equal to of As is the multiplicity of in , cannot divide the value at of the other factors of Thus there is no prime that divides
If is the multiplicative order of modulo , then By definition, If then would divide another factor of and would thus divide 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 be a prime divisor of such that is not be the multiplicative order of modulo . If is the multiplicative order of modulo , then divides both and The resultant of and may be written 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, divides also the discriminant of Thus divides .
and are coprime. In other words, if is a prime common divisor of and 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 and then does not divide Let . It suffices to prove that does not divide for some polynomial , which is a multiple of We take
The multiplicative order of modulo divides , which is a divisor of . Thus is a multiple of . Now,
As is prime and greater than 2, all the terms but the first one are multiples of This proves that
Applications
Using , one can give an elementary proof for the infinitude of primes congruent to 1 modulo n, which is a special case of Dirichlet's theorem on arithmetic progressions.
Suppose is a finite list of primes congruent to modulo Let and consider . Let be a prime factor of (to see that decompose it into linear factors and note that 1 is the closest root of unity to ). Since we know that is a new prime not in the list. We will show that
Let be the order of modulo Since we have . Thus . We will show that .
Assume for contradiction that . Since
we have
for some . Then is a double root of
Thus must be a root of the derivative so
But and therefore This is a contradiction so . The order of which is , must divide . Thus
See also
Cyclotomic field
Aurifeuillean factorization
Root of unity
References
Further reading
Gauss's book Disquisitiones Arithmeticae has been translated from Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.
External links
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Algebra
Number theory