Polynomial connecting together the real part of the roots of unity.
In
number theory, the real parts of the
roots of unity are related to one-another by means of the
minimal polynomial of The roots of the minimal polynomial are twice the real part of the roots of unity, where the real part of a root of unity is just with
coprime with
Formal definition
For an
integer , the
minimal polynomial of is the non-zero
monic polynomial of smallest degree for which .
For every n, the polynomial is monic, has integer coefficients, and is
irreducible over the integers and the
rational numbers. All its
roots are
real; they are the real numbers with
coprime with and either or These roots are twice the
real parts of the
primitive nth roots of unity. The number of integers relatively prime to is given by
Euler's totient function it follows that the
degree of is for and for
The first two polynomials are and
The polynomials are typical examples of irreducible polynomials whose roots are all real and which have a
cyclic
Galois group.
Examples
The first few polynomials are
Explicit form if n is odd
If is an odd prime, the polynomial can be written in terms of binomial coefficients following a "zigzag path" through
Pascal's triangle:
Putting and
then we have for primes .
If is odd but not a prime, the same polynomial , as can be expected, is reducible and, corresponding to the structure of the
cyclotomic polynomials reflected by the formula , turns out to be just the product of all for the divisors of , including itself:
This means that the are exactly the irreducible factors of , which allows to easily obtain for any odd , knowing its degree . For example,
Explicit form if n is even
From the below
formula in terms of Chebyshev polynomials and the product formula for odd above, we can derive for even
Independently of this, if is an even prime power, we have for the recursion (see
OEIS:
A158982)
- ,
starting with .
Roots
The roots of are given by ,
[1] where and . Since is monic, we have
Combining this result with the fact that the
function is
even, we find that is an
algebraic integer for any positive integer and any integer .
Relation to the cyclotomic polynomials
For a positive integer , let , a primitive -th root of unity. Then the minimal polynomial of is given by the -th
cyclotomic polynomial . Since , the relation between and is given by . This relation can be exhibited in the following identity proved by Lehmer, which holds for any non-zero
complex number :
[2]
Relation to Chebyshev polynomials
In 1993, Watkins and Zeitlin established the following relation between and
Chebyshev polynomials of the first kind.
[1]
If is
odd, then[
verification needed]
and if is
even, then
If is a power of , we have moreover directly
[3]
Absolute value of the constant coefficient
The
absolute value of the constant coefficient of can be determined as follows:
[4]
Generated algebraic number field
The
algebraic number field is the maximal real subfield of a
cyclotomic field . If denotes the
ring of integers of , then . In other words, the
set is an integral basis of . In view of this, the
discriminant of the algebraic number field is equal to the
discriminant of the polynomial , that is
[5]
References