Editing Root of unity (section)

==Algebraic expression==

The {{math|''n''}}th roots of unity are, by definition, the [[root of a polynomial|roots]] of the [[polynomial]] {{math|''x<sup>n</sup>'' − 1}}, and are thus [[algebraic number]]s. As this polynomial is not [[irreducible polynomial|irreducible]] (except for {{math|1=''n'' = 1}}), the primitive {{math|''n''}}th roots of unity are roots of an irreducible polynomial (over the integers) of lower degree, called the {{math|''n''}}th [[cyclotomic polynomial]], and often denoted {{math|Φ<sub>''n''</sub>}}. The degree of {{math|Φ<sub>''n''</sub>}} is given by [[Euler's totient function]], which counts (among other things) the number of primitive {{math|''n''}}th roots of unity.<ref name="riesel">{{cite book
|last= Riesel |first= Hans |author-link= Hans Riesel
|year= 1994
|title= Prime Factorization and Computer Methods for Factorization
|url= https://books.google.com/books?id=ITvaBwAAQBAJ&pg=PA306
|page= 306
|publisher= Springer
|isbn= 0-8176-3743-5}}</ref> The roots of {{math|Φ<sub>''n''</sub>}} are exactly the primitive {{math|''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 <math>\sqrt[n]{1}</math> 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 {{mvar|n}}, there exists an expression built from integers by root extractions, additions, subtractions, multiplications, and divisions (and nothing else), such that the primitive {{mvar|n}}th roots of unity are exactly the set of values that can be obtained by choosing values for the root extractions ({{mvar|k}} possible values for a {{mvar|k}}th root). (For more details see {{slink||Cyclotomic fields}}, below.)

Gauss [[mathematical proof|proved]] that a primitive {{mvar|n}}th root of unity can be expressed using only [[square root]]s, addition, subtraction, multiplication and division if and only if it is possible to [[Compass-and-straightedge construction|construct with compass and straightedge]] the [[regular polygon|regular {{mvar|n}}-gon]]. This is the case [[if and only if]] {{math|''n''}} is either a [[power of two]] or the product of a power of two and [[Fermat prime]]s that are all different.

If {{mvar|z}} is a primitive {{mvar|n}}th root of unity, the same is true for {{math|1/''z''}}, and <math>r=z+\frac 1z</math> is twice the real part of {{mvar|z}}. In other words, {{math|Φ<sub>''n''</sub>}} is a [[reciprocal polynomial]], the polynomial <math>R_n</math> that has {{mvar|r}} as a root may be deduced from {{math|Φ<sub>''n''</sub>}} by the standard manipulation on reciprocal polynomials, and the primitive {{mvar|n}}th roots of unity may be deduced from the roots of <math>R_n</math> by solving the [[quadratic equation]] <math>z^2-rz+1=0.</math> That is, the real part of the primitive root is <math>\frac r2,</math> and its imaginary part is <math>\pm i\sqrt{1-\left(\frac r2\right)^2}.</math>

The polynomial <math>R_n</math> is an irreducible polynomial whose roots are all real. Its degree is a power of two, if and only if {{mvar|n}} is a product of a power of two by a product (possibly [[empty product|empty]]) of distinct Fermat primes, and the regular {{mvar|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''.

===Explicit expressions in low degrees===
* For {{math|1=''n'' = 1}}, the cyclotomic polynomial is {{math|Φ<sub>1</sub>(''x'') {{=}} ''x'' − 1}} Therefore, the only primitive first root of unity is 1, which is a non-primitive {{math|''n''}}th root of unity for every ''n'' > 1.
* As {{math|Φ<sub>2</sub>(''x'') {{=}} ''x'' + 1}}, the only primitive second (square) root of unity is −1, which is also a non-primitive {{math|''n''}}th root of unity for every even {{math|''n'' > 2}}. With the preceding case, this completes the list of [[real number|real]] roots of unity.
* As {{math|Φ<sub>3</sub>(''x'') {{=}} ''x''<sup>2</sup> + ''x'' + 1}}, the primitive third ([[cube root|cube]]) roots of unity, which are the roots of this [[quadratic polynomial]], are <math display="block">\frac{-1 + i \sqrt{3}}{2},\ \frac{-1 - i \sqrt{3}}{2} .</math>
* As {{math|Φ<sub>4</sub>(''x'') {{=}} ''x''<sup>2</sup> + 1}}, the two primitive fourth roots of unity are {{math|''i''}} and {{math|−''i''}}.
* As {{math|Φ<sub>5</sub>(''x'') {{=}} ''x''<sup>4</sup> + ''x''<sup>3</sup> + ''x''<sup>2</sup> + ''x'' + 1}}, the four primitive fifth roots of unity are the roots of this [[quartic polynomial]], which may be explicitly solved in terms of radicals, giving the roots <math display="block">\frac{\varepsilon\sqrt 5 - 1}4 \pm i \frac{\sqrt{10 + 2\varepsilon\sqrt 5}}{4},</math> where <math>\varepsilon</math> may take the two values 1 and −1 (the same value in the two occurrences).
* As {{math|Φ<sub>6</sub>(''x'') {{=}} ''x''<sup>2</sup> − ''x'' + 1}}, there are two primitive sixth roots of unity, which are the negatives (and also the square roots) of the two primitive cube roots: <math display="block">\frac{1 + i \sqrt{3}}{2},\ \frac{1 - i \sqrt{3}}{2}.</math>
* As 7 is not a Fermat prime, the seventh roots of unity are the first that require [[cube root]]s. There are 6 primitive seventh roots of unity, which are pairwise [[complex conjugate]]. The sum of a root and its conjugate is twice its real part. These three sums are the three real roots of the cubic polynomial <math>r^3+r^2-2r-1,</math> and the primitive seventh roots of unity are <math display="block">\frac{r}{2}\pm i\sqrt{1-\frac{r^2}{4}},</math> where {{mvar|r}} runs over the roots of the above polynomial. As for every cubic polynomial, these roots may be expressed in terms of square and cube roots. However, as these three roots are all real, this is [[casus irreducibilis]], and any such expression involves non-real cube roots.
* As {{math|Φ<sub>8</sub>(''x'') {{=}} ''x''<sup>4</sup> + 1}}, the four primitive eighth roots of unity are the square roots of the primitive fourth roots, {{math|±&hairsp;''i''}}. They are thus <math display="block"> \pm\frac{\sqrt{2}}{2} \pm i\frac{\sqrt{2}}{2}.</math>
* See [[Heptadecagon]] for the real part of a 17th root of unity.