In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the ''n''th cyclotomic polynomial, for any
positive integer
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
''n'', is the unique
irreducible polynomial with integer coefficients that is a
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
of
and is not a divisor of
for any Its
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
are all ''n''th
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
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
). In other words, the ''n''th cyclotomic polynomial is equal to
:
It may also be defined as the
monic polynomial
In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form:
:x^n+c_x^+\ ...
with integer coefficients that is the
minimal polynomial over the
field of the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s of any
primitive ''n''th-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 ''d''th primitive root of unity for some ''d'' that divides ''n''.
Examples
If ''n'' is a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
, then
:
If ''n'' = 2''p'' where ''p'' is an odd
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
, 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
palindromics of even degree.
The degree of
, or in other words the number of ''n''th primitive roots of unity, is
, where
is
Euler's totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ...
.
The fact that
is an irreducible polynomial of degree
in the
ring