Editing Root of unity (section)

===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.