Root of unity


In mathematics, a root of unity is any complex number that yields 1 when raised to some positive integer power. 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. It is occasionally called a de Moivre number after French mathematician Abraham de Moivre.
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 th roots of unity, except when is a multiple of the characteristic of the field.

General definition

An th root of unity, where is a positive integer, is a number satisfying the equation
Unless otherwise specified, the roots of unity may be taken to be complex numbers, and in this case, the th roots of unity are
However, the defining equation of roots of unity is meaningful over any field , and this allows considering roots of unity in. Whichever is the field, the roots of unity in are either complex numbers, if the characteristic of is 0, or, otherwise, belong to a finite field. Conversely, every nonzero element in a finite field is a root of unity in that field. See Root of unity modulo n and Finite field for further details.
An th root of unity is said to be if it is not an th root of unity for some smaller, that is if
If n is a prime number, then all th roots of unity, except 1, are primitive.
In the above formula in terms of exponential and trigonometric functions, the primitive th roots of unity are those for which and are coprime integers.
Subsequent sections of this article will comply with complex roots of unity. For the case of roots of unity in fields of nonzero characteristic, see. For the case of roots of unity in rings of modular integers, see Root of unity modulo n.

Elementary properties

Every th root of unity is a primitive th root of unity for some, which is the smallest positive integer such that.
Any integer power of an th root of unity is also an th root of unity, as
This is also true for negative exponents. In particular, the reciprocal of an th root of unity is its complex conjugate, and is also an th root of unity:
If is an th root of unity and then. Indeed, by the definition of congruence modulo n, for some integer, and hence
Therefore, given a power of, one has, where is the remainder of the Euclidean division of by.
Let be a primitive th root of unity. Then the powers,, ..., , are th roots of unity and are all distinct. This implies that,, ..., , are all of the th roots of unity, since an th-degree polynomial equation over a field has at most solutions.
From the preceding, it follows that, if is a primitive th root of unity, then if and only if
If is not primitive then implies but the converse may be false, as shown by the following example. If, a non-primitive th root of unity is, and one has, although
Let be a primitive th root of unity. A power of is a primitive th root of unity for
where is the greatest common divisor of and. This results from the fact that is the smallest multiple of that is also a multiple of. In other words, is the least common multiple of and. Thus
Thus, if and are coprime, is also a primitive th root of unity, and therefore there are distinct primitive th roots of unity. This implies that if is a prime number, all the roots except are primitive.
In other words, if is the set of all th roots of unity and is the set of primitive ones, is a disjoint union of the :
where the notation means that goes through all the positive divisors of, including and.
Since the cardinality of is, and that of is, this demonstrates the classical formula

Group properties

Group of all roots of unity

The product and the multiplicative inverse of two roots of unity are also roots of unity. In fact, if and, then, and, where is the least common multiple of and.
Therefore, the roots of unity form an abelian group under multiplication. This group is the torsion subgroup of the circle group.

Group of th roots of unity

For an integer n, the product and the multiplicative inverse of two th roots of unity are also th roots of unity. Therefore, the th roots of unity form an abelian group under multiplication.
Given a primitive th root of unity, the other th roots are powers of. This means that the group of the 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 th roots of unity

Let be the field extension of the rational numbers generated over by a primitive th root of unity. As every th root of unity is a power of, the field contains all th roots of unity, and is a Galois extension of
If is an integer, is a primitive th root of unity if and only if and are coprime. In this case, the map
induces an automorphism of, which maps every th root of unity to its th power. Every automorphism of is obtained in this way, and these automorphisms form the Galois group of over the field of the rationals.
The rules of exponentiation imply that the composition of two such automorphisms is obtained by multiplying the exponents. It follows that the map
defines a group isomorphism between the units of the ring of integers modulo and the Galois group of
This shows that this Galois group is abelian, and implies thus that the primitive roots of unity may be expressed in terms of radicals.

Galois group of the real part of the primitive roots of unity

The real part of the primitive roots of unity are related to one another as roots of the minimal polynomial of The roots of the minimal polynomial are just twice the real part; these roots form a cyclic Galois group.

Trigonometric expression

, which is valid for all real and integers, is
Setting gives a primitive th root of unity – one gets
but
for. In other words,
is a primitive th root of unity.
This formula shows that in the complex plane the th roots of unity are at the vertices of a regular -sided polygon inscribed in the unit circle, with one vertex at 1. This geometric fact accounts for the term "cyclotomic" in such phrases as cyclotomic field and cyclotomic polynomial; it is from the Greek roots "cyclo" plus "tomos".
Euler's formula
which is valid for all real, can be used to put the formula for the th roots of unity into the form
It follows from the discussion in the previous section that this is a primitive th-root if and only if the fraction is in lowest terms; that is, that and are coprime. An irrational number that can be expressed as the real part of the root of unity; that is, as, is called a trigonometric number.

Algebraic expression

The th roots of unity are, by definition, the roots of the polynomial, and are thus algebraic numbers. As this polynomial is not irreducible, the primitive th roots of unity are roots of an irreducible polynomial of lower degree, called the th cyclotomic polynomial, and often denoted. The degree of is given by Euler's totient function, which counts the number of primitive th roots of unity. The roots of are exactly the primitive th roots of unity.
Galois theory can be used to show that the cyclotomic polynomials may be conveniently solved in terms of radicals. This means that, for each positive integer, there exists an expression built from integers by root extractions, additions, subtractions, multiplications, and divisions, such that the primitive th roots of unity are exactly the set of values that can be obtained by choosing values for the root extractions.
Gauss proved that a primitive 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 -gon. This is the case if and only if is either a power of two or the product of a power of two and Fermat primes that are all different.
If is a primitive th root of unity, the same is true for, and is twice the real part of. In other words, is a reciprocal polynomial, the polynomial that has as a root may be deduced from by the standard manipulation on reciprocal polynomials, and the primitive th roots of unity may be deduced from the roots of by solving the quadratic equation That is, the real part of the primitive root is and its imaginary part is
The polynomial is an irreducible polynomial whose roots are all real. Its degree is a power of two, if and only if is a product of a power of two by a product of distinct Fermat primes, and the regular -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, the cyclotomic polynomial is Therefore, the only primitive first root of unity is 1, which is a non-primitive th root of unity for every n > 1.
  • As, the only primitive second root of unity is −1, which is also a non-primitive th root of unity for every even. With the preceding case, this completes the list of real roots of unity.
  • As, the primitive third roots of unity, which are the roots of this quadratic polynomial, are
  • As, the two primitive fourth roots of unity are and.
  • As, 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 where may take the two values 1 and −1.
  • As, there are two primitive sixth roots of unity, which are the negatives of the two primitive cube roots:
  • As 7 is not a Fermat prime, the seventh roots of unity are the first that require cube roots. 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 and the primitive seventh roots of unity are where 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, the four primitive eighth roots of unity are the square roots of the primitive fourth roots,. They are thus
  • See Heptadecagon for the real part of a 17th root of unity.