Minimal polynomial of 2cos(2pi/n)

For an integer formula_1, the minimal polynomial formula_2 of formula_3 is the non-zero monic polynomial of degree formula_4 with integer coefficients, such that formula_5. Here formula_6 denotes the Euler's totient function. In particular, for formula_7 one has formula_8 and formula_9
For every , the polynomial formula_2 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 formula_11 with coprime with and (coprimality implies that can occur only for ). These roots are twice the real parts of the primitive th roots of unity.
The polynomials formula_2 are typical examples of irreducible polynomials whose roots are all real and which have a cyclic Galois group.
Examples.
The first few polynomials formula_2 are
Roots.
The roots of formula_2 are given by formula_16, where formula_17 and formula_18. Since formula_2 is monic, we have
Combining this result with the fact that the function formula_21 is even, we find that formula_16 is an algebraic integer for any positive integer formula_23 and any integer formula_24.
Relation to the cyclotomic polynomials.
For a positive integer formula_23, let formula_26, a primitive formula_23-th root of unity. Then the minimal polynomial of formula_28 is given by the formula_23-th cyclotomic polynomial formula_30. Since formula_31, the relation between formula_3 and formula_28 is given by formula_34. This relation can be exhibited in the following identity proved by Lehmer, which holds for any non-zero complex number formula_35:
Relation to Chebyshev polynomials.
In 1993, Watkins and Zeitlin established the following relation between formula_2 and Chebyshev polynomials of the first kind.
If formula_38 is odd, then
and if formula_40 is even, then
Absolute value of the constant coefficient.
The absolute value of the constant coefficient of formula_2 can be determined as follows:
The algebraic number field formula_44.
The algebraic number field formula_45 is the maximal real subfield of a cyclotomic field formula_46. Various properties of this field were established by Liang. In particular, if we let formula_47 denote the ring of integers of formula_48, then formula_49. In other words, the set formula_50 is an integral basis of formula_47. In view of this, the discriminant of the algebraic number field formula_48 is equal to the discriminant of the polynomial formula_2, and can be computed as follows: