Root of unity

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power $n$ . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform.

Roots of unity can be defined in any field. If the characteristic of the field is zero, the roots are complex numbers that are also algebraic integers. For fields with a positive characteristic, the roots belong to a finite field, and, conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly $n$ $n$ th roots of unity, except when $n$ is a multiple of the (positive) characteristic of the field.

Group properties[edit]

Group of all roots of unity[edit]

The product and the multiplicative inverse of two roots of unity are also roots of unity. In fact, if $x m = 1$ and $y n = 1$ , then $(x -1) m = 1$ , and $(xy) k = 1$ , where $k$ is the least common multiple of $m$ and $n$ .

Therefore, the roots of unity form an abelian group under multiplication. This group is the torsion subgroup of the circle group.

Group of $n$ th roots of unity[edit]

For an integer n, the product and the multiplicative inverse of two $n$ th roots of unity are also $n$ th roots of unity. Therefore, the $n$ th roots of unity form an abelian group under multiplication.

Given a primitive $n$ th root of unity $ω$ , the other $n$ th roots are powers of $ω$ . This means that the group of the $n$ th roots of unity is a cyclic group. It is worth remarking that the term of cyclic group originated from the fact that this group is a subgroup of the circle group.

Galois group of the primitive $n$ th roots of unity[edit]

Let $\mathbb {Q} (\omega )$ be the field extension of the rational numbers generated over $\mathbb {Q}$ by a primitive $n$ th root of unity $ω$ . As every $n$ th root of unity is a power of $ω$ , the field $\mathbb {Q} (\omega )$ contains all $n$ th roots of unity, and $\mathbb {Q} (\omega )$ is a Galois extension of $\mathbb {Q} .$

If $k$ is an integer, $ω k$ is a primitive $n$ th root of unity if and only if $k$ and $n$ are coprime. In this case, the map

For $n = 1$ , the cyclotomic polynomial is $Φ 1 (x) = x - 1$ Therefore, the only primitive first root of unity is 1, which is a non-primitive $n$ th root of unity for every n > 1.

As $Φ 2 (x) = x + 1$ , the only primitive second (square) root of unity is −1, which is also a non-primitive $n$ th root of unity for every even $n > 2$ . With the preceding case, this completes the list of roots of unity.

real

As $Φ 3 (x) = x 2 + x + 1$ , the primitive third () roots of unity, which are the roots of this quadratic polynomial, are ${\frac {-1+i{\sqrt {3}}}{2}},\ {\frac {-1-i{\sqrt {3}}}{2}}.$

cube

As $Φ 4 (x) = x 2 + 1$ , the two primitive fourth roots of unity are $i$ and $- i$ .

As $Φ 5 (x) = x 4 + x 3 + x 2 + x + 1$ , the four primitive fifth roots of unity are the roots of this , which may be explicitly solved in terms of radicals, giving the roots ${\frac {\varepsilon {\sqrt {5}}-1}{4}}\pm i{\frac {\sqrt {10+2\varepsilon {\sqrt {5}}}}{4}},$ where $\varepsilon$ may take the two values 1 and −1 (the same value in the two occurrences).

quartic polynomial

As $Φ 6 (x) = x 2 - x + 1$ , there are two primitive sixth roots of unity, which are the negatives (and also the square roots) of the two primitive cube roots: ${\frac {1+i{\sqrt {3}}}{2}},\ {\frac {1-i{\sqrt {3}}}{2}}.$

As 7 is not a Fermat prime, the seventh roots of unity are the first that require . 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 $r^{3}+r^{2}-2r-1,$ and the primitive seventh roots of unity are ${\frac {r}{2}}\pm i{\sqrt {1-{\frac {r^{2}}{4}}}},$ where $r$ 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.

cube roots

As $Φ 8 (x) = x 4 + 1$ , the four primitive eighth roots of unity are the square roots of the primitive fourth roots, $\pm i$ . They are thus $\pm {\frac {\sqrt {2}}{2}}\pm i{\frac {\sqrt {2}}{2}}.$

See for the real part of a 17th root of unity.

Heptadecagon

Cyclic groups[edit]

The $n$ th roots of unity form under multiplication a cyclic group of order $n$ , and in fact these groups comprise all of the finite subgroups of the multiplicative group of the complex number field. A generator for this cyclic group is a primitive $n$ th root of unity.

The $n$ th roots of unity form an irreducible representation of any cyclic group of order $n$ . The orthogonality relationship also follows from group-theoretic principles as described in Character group.

The roots of unity appear as entries of the eigenvectors of any circulant matrix; that is, matrices that are invariant under cyclic shifts, a fact that also follows from group representation theory as a variant of Bloch's theorem.^[15] In particular, if a circulant Hermitian matrix is considered (for example, a discretized one-dimensional Laplacian with periodic boundaries^[16]), the orthogonality property immediately follows from the usual orthogonality of eigenvectors of Hermitian matrices.

For $n = 3, 6$ they are ( $D = -3$ ).

Eisenstein integers

For $n = 4$ they are ( $D = -1$ ): see Imaginary unit.

Gaussian integers

For $n = 1, 2$ , both roots of unity 1 and −1 are integers.

For three values of $n$ , the roots of unity are quadratic integers:

For four other values of $n$ , the primitive roots of unity are not quadratic integers, but the sum of any root of unity with its complex conjugate (also an $n$ th root of unity) is a quadratic integer.

For $n = 5, 10$ , none of the non-real roots of unity (which satisfy a quartic equation) is a quadratic integer, but the sum $z + z = 2 Re z$ of each root with its complex conjugate (also a 5th root of unity) is an element of the ring $Z [1 + \sqrt 5 / 2]$ ( $D = 5$ ). For two pairs of non-real 5th roots of unity these sums are inverse golden ratio and minus golden ratio.

For $n = 8$ , for any root of unity $z + z$ equals to either 0, ±2, or ±√2 ( $D = 2$ ).

For $n = 12$ , for any root of unity, $z + z$ equals to either 0, ±1, ±2 or ±√3 ( $D = 3$ ).

Argand system

the unit complex numbers

Circle group

Cyclotomic field

Group scheme of roots of unity

Dirichlet character

Ramanujan's sum

Witt vector

Teichmüller character

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