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Minimal polynomial of 2cos(2pi/n)

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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

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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

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The first few polynomials are

Explicit form if n is odd

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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

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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 OEISA158982)

,

starting with .

Roots

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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

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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

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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

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The absolute value of the constant coefficient of can be determined as follows:[4]

Generated algebraic number field

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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

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  1. ^ a b W. Watkins and J. Zeitlin (1993). "The minimal polynomial of ". The American Mathematical Monthly. 100 (5): 471–474. doi:10.2307/2324301. JSTOR 2324301.
  2. ^ D. H. Lehmer (1933). "A note on trigonometric algebraic numbers". The American Mathematical Monthly. 40 (3): 165–166. doi:10.2307/2301023. JSTOR 2301023.
  3. ^ see OEIS A064984
  4. ^ 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.
  5. ^ J. J. Liang (1976). "On the integral basis of the maximal real subfield of a cyclotomic field". Journal für die reine und angewandte Mathematik. 286–287: 223–226.