Root of unity
Algebraic expression

The n th roots of unity are, by definition, the roots of the polynomial x − 1, and are thus algebraic numbers. As this polynomial is not irreducible (except for n = 1), the primitive n th roots of unity are roots of an irreducible polynomial (over the integers) of lower degree, called the n th cyclotomic polynomial, and often denoted Φ. The degree of Φ is given by Euler's totient function, which counts (among other things) the number of primitive n th roots of unity. The roots of Φ are exactly the primitive n th roots of unity.

Galois theory can be used to show that the cyclotomic polynomials may be conveniently solved in terms of radicals. (The trivial form 1 n {\displaystyle {\sqrt{1}}} is not convenient, because it contains non-primitive roots, such as 1, which are not roots of the cyclotomic polynomial, and because it does not give the real and imaginary parts separately.) This means that, for each positive integer n, there exists an expression built from integers by root extractions, additions, subtractions, multiplications, and divisions (and nothing else), such that the primitive n th roots of unity are exactly the set of values that can be obtained by choosing values for the root extractions (k possible values for a k th root). (For more details see § Cyclotomic fields, below.)

Gauss proved that a primitive n th root of unity can be expressed using only square roots, addition, subtraction, multiplication and division if and only if it is possible to construct with compass and straightedge the regular n -gon. This is the case if and only if n is either a power of two or the product of a power of two and Fermat primes that are all different.

If z is a primitive n th root of unity, the same is true for 1/ z, and r = z + 1 z {\displaystyle r=z+{\frac {1}{z}}} is twice the real part of z. In other words, Φ is a reciprocal polynomial, the polynomial R n {\displaystyle R_{n}} that has r as a root may be deduced from Φ by the standard manipulation on reciprocal polynomials, and the primitive n th roots of unity may be deduced from the roots of R n {\displaystyle R_{n}} by solving the quadratic equation z 2 − r z + 1 = 0. {\displaystyle z^{2}-rz+1=0.} That is, the real part of the primitive root is r 2, {\displaystyle {\frac {r}{2}},} and its imaginary part is ± i 1 − (r 2) 2. {\displaystyle \pm i{\sqrt {1-\left({\frac {r}{2}}\right)^{2}}}.}

The polynomial R n {\displaystyle R_{n}} is an irreducible polynomial whose roots are all real. Its degree is a power of two, if and only if n is a product of a power of two by a product (possibly empty) of distinct Fermat primes, and the regular n -gon is constructible with compass and straightedge. Otherwise, it is solvable in radicals, but one are in the casus irreducibilis, that is, every expression of the roots in terms of radicals involves nonreal radicals.
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