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
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.
The first few polynomials
are
![{\displaystyle {\begin{aligned}\Psi _{1}(x)&=x-2\\\Psi _{2}(x)&=x+2\\\Psi _{3}(x)&=x+1\\\Psi _{4}(x)&=x\\\Psi _{5}(x)&=x^{2}+x-1\\\Psi _{6}(x)&=x-1\\\Psi _{7}(x)&=x^{3}+x^{2}-2x-1\\\Psi _{8}(x)&=x^{2}-2\\\Psi _{9}(x)&=x^{3}-3x+1\\\Psi _{10}(x)&=x^{2}-x-1\\\Psi _{11}(x)&=x^{5}+x^{4}-4x^{3}-3x^{2}+3x+1\\\Psi _{12}(x)&=x^{2}-3\\\Psi _{13}(x)&=x^{6}+x^{5}-5x^{4}-4x^{3}+6x^{2}+3x-1\\\Psi _{14}(x)&=x^{3}-x^{2}-2x+1\\\Psi _{15}(x)&=x^{4}-x^{3}-4x^{2}+4x+1\\\Psi _{16}(x)&=x^{4}-4x^{2}+2\\\Psi _{17}(x)&=x^{8}+x^{7}-7x^{6}-6x^{5}+15x^{4}+10x^{3}-10x^{2}-4x+1\\\Psi _{18}(x)&=x^{3}-3x-1\\\Psi _{19}(x)&=x^{9}+x^{8}-8x^{7}-7x^{6}+21x^{5}+15x^{4}-20x^{3}-10x^{2}+5x+1\\\Psi _{20}(x)&=x^{4}-5x^{2}+5\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/8b23662d3e9ecbd27d2aa59d354ae35c1d3d3b22)
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
![{\displaystyle {\begin{aligned}\chi _{n}(x):&={\binom {m}{0}}x^{m}+{\binom {m-1}{0}}x^{m-1}-{\binom {m-1}{1}}x^{m-2}-{\binom {m-2}{1}}x^{m-3}+{\binom {m-2}{2}}x^{m-4}+{\binom {m-3}{2}}x^{m-5}--++\cdots \\&=\sum _{k=0}^{m}(-1)^{\lfloor k/2\rfloor }{\binom {m-\lfloor (k+1)/2\rfloor }{\lfloor k/2\rfloor }}x^{m-k}\\&={\binom {m}{m}}x^{m}+{\binom {m-1}{m-1}}x^{m-1}-{\binom {m-1}{m-2}}x^{m-2}-{\binom {m-2}{m-3}}x^{m-3}+{\binom {m-2}{m-4}}x^{m-4}+{\binom {m-3}{m-5}}x^{m-5}--++\cdots \\&=\sum _{k=0}^{m}(-1)^{\lfloor (m-k)/2\rfloor }{\binom {\lfloor (m+k)/2\rfloor }{k}}x^{k},\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/05b5cadbeb3ed980096ecf1727e59df9f79930b2)
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:
![{\displaystyle \prod _{d\mid n \atop d>1}\Psi _{d}(x)=\chi _{n}(x).}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/0a278137ffd14fbc744c0cdf5de84baa6900b20a)
This means that the
are exactly the irreducible factors of
, which allows to easily obtain
for any odd
, knowing its degree
. For example,
![{\displaystyle {\begin{aligned}\chi _{15}(x)&=x^{7}+x^{6}-6x^{5}-5x^{4}+10x^{3}+6x^{2}-4x-1\\&=(x+1)(x^{2}+x-1)(x^{4}-x^{3}-4x^{2}+4x+1)\\&=\Psi _{3}(x)\cdot \Psi _{5}(x)\cdot \Psi _{15}(x).\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/c9cb7d475488b1c0b5af1f055a93f34beb2b4634)
From the below formula in terms of Chebyshev polynomials and the product formula for odd
above, we can derive for even
![{\displaystyle \prod _{d\mid n \atop d>1}\Psi _{d}(x)={\Big (}\chi _{n+1}(x)+\chi _{n-1}(x){\Big )}.}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/cbeb03c167802d8e7a8f13a417b2a4cad64ee0cf)
Independently of this, if
is an even prime power, we have for
the recursion (see OEIS: A158982)
,
starting with
.
The roots of
are given by
,[1] where
and
. Since
is monic, we have
![{\displaystyle \Psi _{n}(x)=\displaystyle \prod _{\begin{array}{c}1\leq k<{\frac {n}{2}}\\\gcd(k,n)=1\end{array}}\left(x-2\cos \left({\frac {2\pi k}{n}}\right)\right).}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/8aa1d218adf6e38f1438acd60c50b9bdcffe8530)
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
[edit]
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]
![{\displaystyle \Psi _{n}\left(z+z^{-1}\right)=z^{-{\frac {\varphi (n)}{2}}}\Phi _{n}(z)}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/9df1fe655dd97d83b43dca96edf561f1c9932f59)
Relation to Chebyshev polynomials
[edit]
In 1993, Watkins and Zeitlin established the following relation between
and Chebyshev polynomials of the first kind.[1]
If
is odd, then[verification needed]
![{\displaystyle \prod _{d\mid n}\Psi _{d}(2x)=2(T_{s+1}(x)-T_{s}(x)),}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/d803261af3cf5ff5c6c9ff618054ac90be639bea)
and if
is even, then
![{\displaystyle \prod _{d\mid n}\Psi _{d}(2x)=2(T_{s+1}(x)-T_{s-1}(x)).}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/40a836b6eea1aab59c3244a2aa4e7c99a46acb57)
If
is a power of
, we have moreover directly[3]
![{\displaystyle \Psi _{2^{k+1}}(2x)=2T_{2^{k-1}}(x).}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/bc5c7b74bfe8dbc5c5f7d1691c0f4082f3727259)
Absolute value of the constant coefficient
[edit]
The absolute value of the constant coefficient of
can be determined as follows:[4]
![{\displaystyle |\Psi _{n}(0)|={\begin{cases}0&{\text{if}}\ n=4,\\2&{\text{if}}\ n=2^{k},k\geq 0,k\neq 2,\\p&{\text{if}}\ n=4p^{k},k\geq 1,p>2\ {\text{prime,}}\\1&{\text{otherwise.}}\end{cases}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/d804e51620ae11162c3715a6e80d62cb28b7a7c3)
Generated algebraic number field
[edit]
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]
![{\displaystyle D_{K_{n}}={\begin{cases}2^{(m-1)2^{m-2}-1}&{\text{if}}\ n=2^{m},m>2,\\p^{(mp^{m}-(m+1)p^{m-1}-1)/2}&{\text{if}}\ n=p^{m}\ {\text{or}}\ 2p^{m},p>2\ {\text{prime}},\\\left(\prod _{i=1}^{\omega (n)}p_{i}^{e_{i}-1/(p_{i}-1)}\right)^{\frac {\varphi (n)}{2}}&{\text{if}}\ \omega (n)>1,k\neq 2p^{m}.\end{cases}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/9c1c69cba56df8fbb7388f0cf7ed8f4453ecef4d)