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
The polynomials are typical examples of irreducible polynomials whose roots are all real and which have a cyclicGalois group.
Examplesedit
The first few polynomials are
Explicit form if n is oddedit
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,
Independently of this, if is an even prime power, we have for the recursion (see OEIS: A158982)
,
starting with .
Rootsedit
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 polynomialsedit
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 polynomialsedit
In 1993, Watkins and Zeitlin established the following relation between and Chebyshev polynomials of the first kind.[1]
^C. Adiga, I. N. Cangul and H. N. Ramaswamy (2016). "On the constant term of the minimal polynomial of over ". Filomat. 30 (4): 1097–1102. doi:10.2298/FIL1604097A.