A polynomial given by
(1)

where are the roots of unity in given by
(2)

and runs over integers relatively prime to . The prime may be dropped if the product is instead taken over primitive roots of unity, so that
(3)

The notation is also frequently encountered. Dickson et al. (1923) and Apostol (1975) give extensive bibliographies for cyclotomic polynomials.
The cyclotomic polynomial for can also be defined as
(4)

where is the Möbius function and the product is taken over the divisors of (Vardi 1991, p. 225).
is an integer polynomial and an irreducible polynomial with polynomial degree , where is the totient function. Cyclotomic polynomials are returned by the Wolfram Language command Cyclotomic[n, x]. The roots of cyclotomic polynomials lie on the unit circle in the complex plane, as illustrated above for the first few cyclotomic polynomials.
The first few cyclotomic polynomials are
(5)
 
(6)
 
(7)
 
(8)
 
(9)
 
(10)
 
(11)
 
(12)
 
(13)
 
(14)

The cyclotomic polynomial is illustrated above in the complex plane.
On any line through the origin, the value of a cyclotomic polynomial is strictly increasing outside the unit disk.
If is an odd prime, then
(15)
 
(16)
 
(17)
 
(18)
 
(19)
 
(20)

(Riesel 1994, p. 306). Similarly, for again an odd prime,
(21)
 
(22)
 
(23)

For the first few remaining values of ,
(24)
 
(25)
 
(26)
 
(27)
 
(28)
 
(29)
 
(30)
 
(31)

(Riesel 1994, p. 307).
For a prime relatively prime to ,
(32)

but if ,
(33)

(Nagell 1951, p. 160).
An explicit equation for for squarefree is given by
(34)

where is the totient function and is calculated using the recurrence relation
(35)

with , where is the Möbius function and is the greatest common divisor of and .
The polynomial can be factored as
(36)

Furthermore,
(37)
 
(38)

The coefficients of the inverse of the cyclotomic polynomial
(39)
 
(40)

can also be computed from
(41)
 
(42)
 
(43)

where is the floor function.
For prime,
(44)

i.e., the coefficients are all 1. The first cyclotomic polynomial to have a coefficient other than and 0 is , which has coefficients of for and . This is true because 105 is the first number to have three distinct odd prime factors, i.e., (McClellan and Rader 1979, Schroeder 1997). The smallest values of for which has one or more coefficients , , , ... are 0, 105, 385, 1365, 1785, 2805, 3135, 6545, 6545, 10465, 10465, 10465, 10465, 10465, 11305, ... (OEIS A013594).
It appears to be true that, for , if factors, then the factors contain a cyclotomic polynomial. For example,
(45)
 
(46)

This observation has been checked up to (Nicol 2000). If and are prime, then is irreducible.
Migotti (1883) showed that coefficients of for and distinct primes can be only 0, . Lam and Leung (1996) considered
(47)

for prime. Write the totient function as
(48)

and let
(49)

then
1. iff for some and ,
2. iff for and ,
3. otherwise .
The number of terms having is , and the number of terms having is . Furthermore, assume , then the middle coefficient of is .
Resultants of cyclotomic polynomials have been computed by Lehmer (1936), Diederichsen (1940), and Apostol (1970). It is known that if , i.e., and are relatively prime (Apostol 1975). Apostol (1975) showed that for positive integers and and arbitrary nonzero complex numbers and ,
(50)

where is the greatest common divisor of and , is the totient function, is the Möbius function, and the product is over the divisors of . If and are distinct primes and , then (50) simplifies to
(51)

The following table gives the resultants , (OEIS A054372).
1  2  3  4  5  6  7  
1  0  
2  2  0  
3  3  1  0  
4  2  2  1  0  
5  5  1  1  1  0  
6  1  3  4  1  1  0  
7  7  1  1  1  1  1  0 
The numbers of 1s in successive rows of this table are given by 0, 0, 1, 1, 3, 3, 5, 4, 6, 7, 9, ... (OEIS A075795).
The cyclotomic polynomial has the particularly nice Maclaurin series
(52)

whose coefficients 1, 0, , , 0, 1, 1, 0, , , ... (OEIS A010892) are given by solving the recurrence equation
(53)

with (Wolfram 2002, p. 128), giving the explicit form
(54)

Interestingly, any sequence satisfying the linear recurrence equation
(55)

can be written as
(56)
