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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 k < n. Its roots are all nth primitive roots of unity , where k runs over the positive integers not greater than n and coprime to n. 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 x is a root of if and only if it is a dth primitive root of unity for some d that divides n.

Contents

ExamplesEdit

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:[1]

 

The case of the 105th cyclotomic polynomial is interesting because 105 is the lowest 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:

 

PropertiesEdit

Fundamental toolsEdit

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 palindromics 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.[2] 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 Fourier transform of functions of the greatest common divisor together with the Möbius inversion formula gives:[3]

 

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 allows computation of   on a computer for any n, as soon as integer factorization and division of polynomials are available. Many computer algebra systems have a built in function to compute the cyclotomic polynomials. For example, this function is called by typing cyclotomic_polynomial(n,x) in SageMath, numtheory[cyclotomic](n,x); in Maple, and Cyclotomic[n,x] in Mathematica.

Easy cases for computationEdit

As noted above, if n is a prime number, then

 

If n is an odd integer greater than one, then

 

In particular, if n = 2p is twice an odd prime, then (as noted above)

 

If n = pm is a prime power (where p is prime), then

 

More generally, if n = pmr with r relatively prime to p, 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 q is the product of the prime divisors of n (its radical), then[4]

 

This allows to give formulas for the nth cyclotomic polynomial when n has at most one odd prime factor: If p is an odd prime number, and h and k are positive integers, then:

 
 
 

For the other values of n, the computation of the nth cyclotomic polynomial is similarly reduced to that of   where q is the product of the distinct odd prime divisors of n. To deal with this case, one has that, for p prime and not dividing n,[5]

 

Integers appearing as coefficientsEdit

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 has at most two distinct odd prime factors, then Migotti showed that the coefficients of   are all in the set {1, −1, 0}.[6]

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 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.[7]

Gauss's formulaEdit

Let n be odd, square-free, and greater than 3. Then:[8][9]

 

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 formulaEdit

Let n be odd, square-free and greater than 3. Then[10]

 

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:

 

Cyclotomic polynomials over a finite field and over p-adic integersEdit

Over a finite field with a prime number p of elements, for any integer n that is not a multiple of p, the cyclotomic polynomial   factorizes into   irreducible polynomials of degree d, where   is Euler's totient function, and d is the multiplicative order of p modulo n. In particular,   is irreducible if and only if p is a primitive root modulo n, that is, p does not divide n, and its multiplicative order modulo n is   the degree of  .[citation needed]

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

Polynomial valuesEdit

If x takes any real value, then   for every n ≥ 3 (this follows from the fact that the roots of a cyclotomic polynomial are all non-real, for n ≥ 3).

For studying the values that a cyclotomic polynomial may take when x is given an integer value, it suffices to consider only the case n ≥ 3, as the cases n = 1 and n = 2 are trivial (one has   and  ).

For n ≥ 2, one has

 
  if n is not a prime power,
  if   is a prime power with k ≥ 1.

The values that a cyclotomic polynomial   may take for other integer values of x is strongly related with the multiplicative order modulo a prime number.

More precisely, given a prime number p and an integer b coprime with p, the multiplicative order of b modulo p, is the smallest positive integer n such that p is a divisor of   For b > 1, the multiplicative order of b modulo p is also the shortest period of the representation of 1/p in the numeral base b (see Unique prime; this explains the notation choice).

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

If n > 0 is a positive integer and b > 1 is an integer, then (see below for a proof)

 

where

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

This implies that, if p is an odd prime divisor of   then either n is a divisor of p − 1 or p is a divisor of n. In the latter case   does not divides  

Zsigmondy's theorem implies that the only cases where b > 1 and h = 1 are

 

It follows from above factorization that the odd prime factors of

 

are exactly the odd primes p such that n is the multiplicative order of b modulo p. This fraction may be even only when b is odd. In this case, the multiplicative order of b modulo 2 is always 1.

There are many pairs (n, b) with b > 1 such that   is prime. In fact, Bunyakovsky conjecture implies that, for every n, there are infinitely many b > 1 such that   is prime. See OEISA085398 for the list of the smallest b > 1 such that   is prime (the smallest b > 1 such that   is prime is about  , where   is Euler–Mascheroni constant, and   is Euler's totient function). See also OEISA206864 for the list of the smallest primes of the form   with n > 2 and b > 1, and, more generally, OEISA206942, for the smallest positive integers of this form.

Proofs
  • Values of   If   is a prime power, then
 
If n is not a prime power, let   we have   and P is the product of the   for k dividing n and different of 1. If p is a prime divisor of multiplicity m in n, then   divide P(x), and their values at 1 are m factors equal to p of   As m is the multiplicity of p in n, p cannot divide the value at 1 of the other factors of   Thus there is no prime that divides  
  • If n is the multiplicative order of b modulo p, then   By definition,   If   then p would divide another factor   of   and would thus divide   showing that, if there would be the case, n would not be the multiplicative order of b modulo p.
  • The other prime divisors of   are divisors of n. Let p be a prime divisor of   such that n is not be the multiplicative order of b modulo p. If k is the multiplicative order of b modulo p, then p divides both   and   The resultant of   and   may be written   where P and Q are polynomials. Thus p divides this resultant. As k divides n, and the resultant of two polynomials divides the discriminant of any common multiple of these polynomials, p divides also the discriminant   of   Thus p divides n.
  • g and h are coprime. In other words, if p is a prime common divisor of n and   then n is not the multiplicative order of b modulo p. By Fermat's little theorem, the multiplicative order of b is a divisor of p − 1, and thus smaller than n.
  • g is square-free. In other words, if p is a prime common divisor of n and   then   does not divide   Let n = pm. It suffices to prove that   does not divides S(b) for some polynomial S(x), which is a multiple of   We take
 
The multiplicative order of b modulo p divides gcd(n, p − 1), which is a divisor of m = n/p. Thus c = bm − 1 is a multiple of p. Now,
 
As p is prime and greater than 2, all the terms but the first one are multiples of   This proves that  

ApplicationsEdit

Using  , one can give an elementary proof for the infinitude of primes congruent to 1 modulo n,[11] which is a special case of Dirichlet's theorem on arithmetic progressions.

Proof

Suppose   are 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 alsoEdit

NotesEdit

  1. ^ Sloane, N. J. A. (ed.). "Sequence A013595". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  2. ^ Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556
  3. ^ Schramm, Wolfgang (2015). "Eine alternative Produktdarstellung für die Kreisteilungspolynome". Elemente der Mathematik. Swiss Mathematical Society. 70 (4): 137–143. Archived from the original on 2015-12-22. Retrieved 2015-10-10.
  4. ^ Cox, David A. (2012), "Exercise 12", Galois Theory (2nd ed.), John Wiley & Sons, p. 237, doi:10.1002/9781118218457, ISBN 978-1-118-07205-9.
  5. ^ Weisstein, Eric W. "Cyclotomic Polynomial". Retrieved 12 March 2014.
  6. ^ Isaacs, Martin (2009). Algebra: A Graduate Course. AMS Bookstore. p. 310. ISBN 978-0-8218-4799-2.
  7. ^ Meier (2008)
  8. ^ Gauss, DA, Articles 356-357
  9. ^ Riesel, pp. 315-316, p. 436
  10. ^ Riesel, pp. 309-315, p. 443
  11. ^ S. Shirali. Number Theory. Orient Blackswan, 2004. p. 67. ISBN 81-7371-454-1

ReferencesEdit

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.

  • Gauss, Carl Friedrich (1986) [1801]. Disquisitiones Arithmeticae. Translated into English by Clarke, Arthur A. (2nd corr. ed.). New York: Springer. ISBN 0387962549.
  • Gauss, Carl Friedrich (1965) [1801]. Untersuchungen uber hohere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory). Translated into German by Maser, H. (2nd ed.). New York: Chelsea. ISBN 0-8284-0191-8.
  • Lemmermeyer, Franz (2000). Reciprocity Laws: from Euler to Eisenstein. Berlin: Springer. doi:10.1007/978-3-662-12893-0. ISBN 978-3-642-08628-1.
  • Maier, Helmut (2008), "Anatomy of integers and cyclotomic polynomials", in De Koninck, Jean-Marie; Granville, Andrew; Luca, Florian (eds.), Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13-17, 2006, CRM Proceedings and Lecture Notes, 46, Providence, RI: American Mathematical Society, pp. 89–95, ISBN 978-0-8218-4406-9, Zbl 1186.11010
  • Riesel, Hans (1994). Prime Numbers and Computer Methods for Factorization (2nd ed.). Boston: Birkhäuser. ISBN 0-8176-3743-5.

External linksEdit