Root of unity#General definition

File:visualisation_complex_number_roots.svg

An {{mvar|n}}th root of unity, where {{mvar|n}} is a positive integer, is a number {{mvar|z}} satisfying the equation{{Cite book|author=Hadlock, Charles R.|author-link=Charles Robert Hadlock|title=Field Theory and Its Classical Problems, Volume 14|publisher=Cambridge University Press|year=2000|isbn=978-0-88385-032-9|pages=84–86|url=https://books.google.com/books?id=5s1p0CyafnEC&pg=PA84}}{{cite book|last = Lang|first = Serge|chapter=Roots of unity|title=Algebra|publisher=Springer|year=2002|isbn=978-0-387-95385-4|pages=276–277|chapter-url=https://books.google.com/books?id=Fge-BwqhqIYC&pg=PA276}}

$$z^n\; =\; 1.$$

Unless otherwise specified, the roots of unity may be taken to be complex numbers (including the number 1, and the number −1 if {{mvar|n}} is even, which are complex with a zero imaginary part), and in this case, the {{mvar|n}}th roots of unity are{{cite book

|last = Meserve |first = Bruce E.

|title = Fundamental Concepts of Algebra

|page = 52

|publisher = Dover Publications

|year = 1982}}

$$\backslash exp\backslash left(\backslash frac\{2k\backslash pi\; i\}\{n\}\backslash right)=\backslash cos\backslash frac\{2k\backslash pi\}\{n\}+i\backslash sin\backslash frac\{2k\backslash pi\}\{n\},\backslash qquad\; k=0,1,\backslash dots,\; n-1.$$

However, the defining equation of roots of unity is meaningful over any field (and even over any ring) {{math|F}}, and this allows considering roots of unity in {{math|F}}. Whichever is the field {{math|F}}, the roots of unity in {{math|F}} are either complex numbers, if the characteristic of {{math|F}} 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 {{mvar|n}}th root of unity is said to be {{visible anchor|primitive}} if it is not an {{mvar|m}}th root of unity for some smaller {{mvar|m}}, that is if{{cite book

|last = Moskowitz |first= Martin A.

|year = 2003

|title = Adventure in Mathematics

|publisher = World Scientific

|url = https://books.google.com/books?id=YT2_Kqsnn9wC&pg=PA36

|page = 36|isbn= 9789812794949

}}{{cite book

|last1 = Lidl |first1 = Rudolf

|last2 = Pilz |first2 = Günter |author-link2 = Günter Pilz

|year = 1984

|title = Applied Abstract Algebra

|series = Undergraduate Texts in Mathematics

|url = https://books.google.com/books?id=irXSBwAAQBAJ&pg=PA149

|page = 149

|publisher = Springer

|doi = 10.1007/978-1-4615-6465-2

|isbn = 978-0-387-96166-8

}}

:$z^n=1\backslash quad\; \backslash text\{and\}\; \backslash quad\; z^m\; \backslash ne\; 1\; \backslash text\{\; for\; \}\; m\; =\; 1,\; 2,\; 3,\; \backslash ldots,\; n-1.$

If n is a prime number, then all {{math|n}}th roots of unity, except 1, are primitive.{{cite book

|last = Morandi | first = Patrick

|title = Field and Galois theory

| series = Graduate Texts in Mathematics

|year = 1996

| volume = 167

|url = https://books.google.com/books?id=jQ7c8Xqpqk0C&pg=PA74

|page = 74

|publisher = Springer

|isbn = 978-0-387-94753-2

|doi = 10.1007/978-1-4612-4040-2}}

In the above formula in terms of exponential and trigonometric functions, the primitive {{mvar|n}}th roots of unity are those for which {{mvar|k}} and {{mvar|n}} 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 {{slink|Finite field|Roots of unity}}. For the case of roots of unity in rings of modular integers, see Root of unity modulo n.

Every {{math|n}}th root of unity {{math|z}} is a primitive {{math|a}}th root of unity for some {{math|a ≤ n}}, which is the smallest positive integer such that {{math|1=z^a = 1}}.

Any integer power of an {{math|n}}th root of unity is also an {{math|n}}th root of unity,{{cite book

|last = Reilly |first= Norman R.

|year = 2009

|title = Introduction to Applied Algebraic Systems

|url = https://books.google.com/books?id=q33he4hOlKcC&pg=PA137

|page = 137

|publisher= Oxford University Press

|isbn = 978-0-19-536787-4

}} as

:$(z^k)^n\; =\; z^\{kn\}\; =\; (z^n)^k\; =\; 1^k\; =\; 1.$

This is also true for negative exponents. In particular, the reciprocal of an {{math|n}}th root of unity is its complex conjugate, and is also an {{math|n}}th root of unity:{{cite book

|last = Rotman|first = Joseph J.|author-link = Joseph J. Rotman

|title = Advanced Modern Algebra

|year = 2015

|edition = 3rd

|volume = 1

|url = https://books.google.com/books?id=SugUCwAAQBAJ&pg=PA129

|page = 129

|publisher = American Mathematical Society

| isbn=9781470415549 }}

:$\backslash frac\{1\}\{z\}\; =\; z^\{-1\}\; =\; z^\{n-1\}\; =\; \backslash bar\; z.$

If {{math|z}} is an {{math|n}}th root of unity and {{math|a ≡ b (mod n)}} then {{math|1=z^a = z^b}}. Indeed, by the definition of congruence modulo n, {{math|1=a = b + kn}} for some integer {{math|k}}, and hence

:$z^a\; =\; z^\{b+kn\}\; =\; z^b\; z^\{kn\}\; =\; z^b\; (z^n)^k\; =\; z^b\; 1^k\; =\; z^b.$

Therefore, given a power {{math|z^a}} of {{math|z}}, one has {{math|1=z^a = z^r}}, where {{math|0 ≤ r < n}} is the remainder of the Euclidean division of {{mvar|a}} by {{mvar|n}}.

Let {{math|z}} be a primitive {{math|n}}th root of unity. Then the powers {{math|z}}, {{math|z²}}, ..., {{math|zⁿ⁻¹}}, {{math|1=zⁿ = z⁰ = 1}} are {{math|n}}th roots of unity and are all distinct. (If {{math|1=z^a = z^b}} where {{math|1 ≤ a < b ≤ n}}, then {{math|1=z^b−a = 1}}, which would imply that {{math|z}} would not be primitive.) This implies that {{math|z}}, {{math|z²}}, ..., {{math|zⁿ⁻¹}}, {{math|1=zⁿ = z⁰ = 1}} are all of the {{math|n}}th roots of unity, since an {{math|n}}th-degree polynomial equation over a field (in this case the field of complex numbers) has at most {{math|n}} solutions.

From the preceding, it follows that, if {{math|z}} is a primitive {{math|n}}th root of unity, then $z^a\; =\; z^b$ if and only if $a\backslash equiv\; b\; \backslash pmod\{\; n\}.$

If {{math|z}} is not primitive then $a\backslash equiv\; b\; \backslash pmod\{\; n\}$ implies $z^a\; =\; z^b,$ but the converse may be false, as shown by the following example. If {{math|1=n = 4}}, a non-primitive {{math|n}}th root of unity is {{math|1= z = −1}}, and one has $z^2\; =\; z^4\; =\; 1$, although $2\; \backslash not\backslash equiv\; 4\; \backslash pmod\{4\}.$

Let {{math|z}} be a primitive {{math|n}}th root of unity. A power {{math|1=w = z^k}} of {{mvar|z}} is a primitive {{math|a}}th root of unity for

:$a\; =\; \backslash frac\{n\}\{\backslash gcd(k,n)\},$

where $\backslash gcd(k,n)$ is the greatest common divisor of {{mvar|n}} and {{mvar|k}}. This results from the fact that {{math|ka}} is the smallest multiple of {{mvar|k}} that is also a multiple of {{mvar|n}}. In other words, {{math|ka}} is the least common multiple of {{mvar|k}} and {{mvar|n}}. Thus

:$a\; =\backslash frac\{\backslash operatorname\{lcm\}(k,n)\}\{k\}=\backslash frac\{kn\}\{k\backslash gcd(k,n)\}=\backslash frac\{n\}\{\backslash gcd(k,n)\}.$

Thus, if {{math|k}} and {{math|n}} are coprime, {{math|z^k}} is also a primitive {{math|n}}th root of unity, and therefore there are {{math|φ(n)}} distinct primitive {{math|n}}th roots of unity (where {{math|φ}} is Euler's totient function). This implies that if {{math|n}} is a prime number, all the roots except {{math|+1}} are primitive.

In other words, if {{math|R(n)}} is the set of all {{math|n}}th roots of unity and {{math|P(n)}} is the set of primitive ones, {{math|R(n)}} is a disjoint union of the {{math|P(n)}}:

:$\backslash operatorname\{R\}(n)\; =\; \backslash bigcup\_\{d\; \backslash ,|\backslash ,\; n\}\backslash operatorname\{P\}(d),$

where the notation means that {{math|d}} goes through all the positive divisors of {{math|n}}, including {{math|1}} and {{math|n}}.

Since the cardinality of {{math|R(n)}} is {{math|n}}, and that of {{math|P(n)}} is {{math|φ(n)}}, this demonstrates the classical formula

:$\backslash sum\_\{d\; \backslash ,|\backslash ,\; n\}\backslash varphi(d)\; =\; n.$

The product and the multiplicative inverse of two roots of unity are also roots of unity. In fact, if {{math|x^m {{=}} 1}} and {{math|yⁿ {{=}} 1}}, then {{math|(x⁻¹){{sup|m}} {{=}} 1}}, and {{math|(xy){{sup|k}} {{=}} 1}}, where {{math|k}} is the least common multiple of {{math|m}} and {{math|n}}.

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

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

Given a primitive {{math|n}}th root of unity {{math|ω}}, the other {{math|n}}th roots are powers of {{math|ω}}. This means that the group of the {{math|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.

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

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

:$\backslash omega\; \backslash mapsto\; \backslash omega^k$

induces an automorphism of $\backslash Q(\backslash omega)$, which maps every {{math|n}}th root of unity to its {{math|k}}th power. Every automorphism of $\backslash Q(\backslash omega)$ is obtained in this way, and these automorphisms form the Galois group of $\backslash Q(\backslash omega)$ 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

:$k\backslash mapsto\; \backslash left(\backslash omega\; \backslash mapsto\; \backslash omega^k\backslash right)$

defines a group isomorphism between the units of the ring of integers modulo n and the Galois group of $\backslash Q(\backslash omega).$

This shows that this Galois group is abelian, and implies thus that the primitive roots of unity may be expressed in terms of radicals.

{{main|Minimal polynomial of 2cos(2pi/n)}}

The real part of the primitive roots of unity are related to one another as roots of the minimal polynomial of $2\backslash cos(2\backslash pi/n).$ The roots of the minimal polynomial are just twice the real part; these roots form a cyclic Galois group.

Image:3rd roots of unity correction.svg

De Moivre's formula, which is valid for all real {{mvar|x}} and integers {{mvar|n}}, is

:$\backslash left(\backslash cos\; x\; +\; i\; \backslash sin\; x\backslash right)^n\; =\; \backslash cos\; nx\; +\; i\; \backslash sin\; nx.$

Setting {{math|1=x = {{sfrac|2π|n}}}} gives a primitive {{mvar|n}}th root of unity – one gets

:$\backslash left(\backslash cos\backslash frac\{2\backslash pi\}\{n\}\; +\; i\; \backslash sin\backslash frac\{2\backslash pi\}\{n\}\backslash right)^\{\backslash !n\}\; =\; \backslash cos\; 2\backslash pi\; +\; i\; \backslash sin\; 2\backslash pi\; =\; 1,$

but

:$\backslash left(\backslash cos\backslash frac\{2\backslash pi\}\{n\}\; +\; i\; \backslash sin\backslash frac\{2\backslash pi\}\{n\}\backslash right)^\{\backslash !k\}\; =\; \backslash cos\backslash frac\{2k\backslash pi\}\{n\}\; +\; i\; \backslash sin\backslash frac\{2k\backslash pi\}\{n\}\; \backslash neq\; 1$

for {{math|1=k = 1, 2, …, n − 1}}. In other words,

:$\backslash cos\backslash frac\{2\backslash pi\}\{n\}\; +\; i\; \backslash sin\backslash frac\{2\backslash pi\}\{n\}$

is a primitive {{mvar|n}}th root of unity.

This formula shows that in the complex plane the {{mvar|n}}th roots of unity are at the vertices of a regular polygon inscribed in the unit circle, with one vertex at 1 (see the plot for {{math|1=n = 3}} on the right). This geometric fact accounts for the term "cyclotomic" in such phrases as cyclotomic field and cyclotomic polynomial; it is from the Greek roots "cyclo" (circle) plus "tomos" (cut, divide).

Euler's formula

:$e^\{i\; x\}\; =\; \backslash cos\; x\; +\; i\; \backslash sin\; x,$

which is valid for all real {{mvar|x}}, can be used to put the formula for the {{mvar|n}}th roots of unity into the form

:

It follows from the discussion in the previous section that this is a primitive {{mvar|n}}th-root if and only if the fraction {{math|{{sfrac|k|n}}}} is in lowest terms; that is, that {{mvar|k}} and {{mvar|n}} are coprime. An irrational number that can be expressed as the real part of the root of unity; that is, as $\backslash cos(2\backslash pi\; k/n)$, is called a trigonometric number.

The {{math|n}}th roots of unity are, by definition, the roots of the polynomial {{math|xⁿ − 1}}, and are thus algebraic numbers. As this polynomial is not 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_n}}. The degree of {{math|Φ_n}} is given by Euler's totient function, which counts (among other things) the number of primitive {{math|n}}th roots of unity.{{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}} The roots of {{math|Φ_n}} 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 $\backslash sqrt[n]\{1\}$ 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 {{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 proved that a primitive {{mvar|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 polygon. 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 primes 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 $r=z+\backslash frac\; 1z$ is twice the real part of {{mvar|z}}. In other words, {{math|Φ_n}} is a reciprocal polynomial, the polynomial $R\_n$ that has {{mvar|r}} as a root may be deduced from {{math|Φ_n}} by the standard manipulation on reciprocal polynomials, and the primitive {{mvar|n}}th roots of unity may be deduced from the roots of $R\_n$ by solving the quadratic equation $z^2-rz+1=0.$ That is, the real part of the primitive root is $\backslash frac\; r2,$ and its imaginary part is $\backslash pm\; i\backslash sqrt\{1-\backslash left(\backslash frac\; r2\backslash right)^2\}.$

The polynomial $R\_n$ 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) 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.

• For {{math|1=n = 1}}, the cyclotomic polynomial is {{math|Φ₁(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|Φ₂(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 roots of unity.
• As {{math|Φ₃(x) {{=}} x² + x + 1}}, the primitive third (cube) roots of unity, which are the roots of this quadratic polynomial, are $$\backslash frac\{-1\; +\; i\; \backslash sqrt\{3\}\}\{2\},\backslash \; \backslash frac\{-1\; -\; i\; \backslash sqrt\{3\}\}\{2\}\; .$$
• As {{math|Φ₄(x) {{=}} x² + 1}}, the two primitive fourth roots of unity are {{math|i}} and {{math|−i}}.
• As {{math|Φ₅(x) {{=}} x⁴ + x³ + x² + 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 $$\backslash frac\{\backslash varepsilon\backslash sqrt\; 5\; -\; 1\}4\; \backslash pm\; i\; \backslash frac\{\backslash sqrt\{10\; +\; 2\backslash varepsilon\backslash sqrt\; 5\}\}\{4\},$$ where $\backslash varepsilon$ may take the two values 1 and −1 (the same value in the two occurrences).
• As {{math|Φ₆(x) {{=}} x² − 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: $$\backslash frac\{1\; +\; i\; \backslash sqrt\{3\}\}\{2\},\backslash \; \backslash frac\{1\; -\; i\; \backslash sqrt\{3\}\}\{2\}.$$
• 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 $r^3+r^2-2r-1,$ and the primitive seventh roots of unity are $$\backslash frac\{r\}\{2\}\backslash pm\; i\backslash sqrt\{1-\backslash frac\{r^2\}\{4\}\},$$ 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|Φ₈(x) {{=}} x⁴ + 1}}, the four primitive eighth roots of unity are the square roots of the primitive fourth roots, {{math|± i}}. They are thus $$\backslash pm\backslash frac\{\backslash sqrt\{2\}\}\{2\}\; \backslash pm\; i\backslash frac\{\backslash sqrt\{2\}\}\{2\}.$$
• See Heptadecagon for the real part of a 17th root of unity.

If {{mvar|z}} is a primitive {{mvar|n}}th root of unity, then the sequence of powers

: {{math|… , z⁻¹, z⁰, z¹, …}}

is {{mvar|n}}-periodic (because {{math|1=z ^j + n = z ^jz ⁿ = z ^j}} for all values of {{mvar|j}}), and the {{mvar|n}} sequences of powers

:{{math|s_k: … , z ^k⋅(−1), z ^k⋅0, z ^k⋅1, …}}

for {{math|1=k = 1, … , n}} are all {{mvar|n}}-periodic (because {{math|1=z ^{k⋅(j + n)} = z ^k⋅j}}). Furthermore, the set {{math|{s₁, … , s_n}}} of these sequences is a basis of the linear space of all {{mvar|n}}-periodic sequences. This means that any {{mvar|n}}-periodic sequence of complex numbers

: {{math|… , x₋₁ , x₀ , x₁, …}}

can be expressed as a linear combination of powers of a primitive {{mvar|n}}th root of unity:

:$x\_j\; =\; \backslash sum\_k\; X\_k\; \backslash cdot\; z^\{k\; \backslash cdot\; j\}\; =\; X\_1\; z^\{1\backslash cdot\; j\}\; +\; \backslash cdots\; +\; X\_n\; \backslash cdot\; z^\{n\; \backslash cdot\; j\}$

for some complex numbers {{math|1=X₁, … , X_n}} and every integer {{mvar|j}}.

This is a form of Fourier analysis. If {{mvar|j}} is a (discrete) time variable, then {{mvar|k}} is a frequency and {{math|X_k}} is a complex amplitude.

Choosing for the primitive {{mvar|n}}th root of unity

:$z\; =\; e^\backslash frac\{2\backslash pi\; i\}\{n\}\; =\; \backslash cos\backslash frac\{2\backslash pi\}\{n\}\; +\; i\; \backslash sin\backslash frac\{2\backslash pi\}\{n\}$

allows {{math|x_j}} to be expressed as a linear combination of {{math|cos}} and {{math|sin}}:

:$x\_j\; =\; \backslash sum\_k\; A\_k\; \backslash cos\; \backslash frac\{2\backslash pi\; jk\}\{n\}\; +\; \backslash sum\_k\; B\_k\; \backslash sin\; \backslash frac\{2\backslash pi\; jk\}\{n\}.$

This is a discrete Fourier transform.

Let {{math|SR(n)}} be the sum of all the {{mvar|n}}th roots of unity, primitive or not. Then

:$\backslash operatorname\{SR\}(n)\; =$

\begin{cases}

1, & n=1\\

0, & n>1.

\end{cases}

This is an immediate consequence of Vieta's formulas. In fact, the {{mvar|n}}th roots of unity being the roots of the polynomial {{math|Xⁿ − 1}}, their sum is the coefficient of degree {{math|n − 1}}, which is either 1 or 0 according whether {{math|1=n = 1}} or {{math|n > 1}}.

Alternatively, for {{math|1=n = 1}} there is nothing to prove, and for {{math|1=n > 1}} there exists a root {{math|z ≠ 1}} – since the set {{math|S}} of all the {{mvar|n}}th roots of unity is a group, {{math|1=z{{space|hair}}S = S}}, so the sum satisfies {{math|1= z SR(n) = SR(n)}}, whence {{math|1=SR(n) = 0}}.

Let {{math|SP(n)}} be the sum of all the primitive {{mvar|n}}th roots of unity. Then

:$\backslash operatorname\{SP\}(n)\; =\; \backslash mu(n),$

where {{math|μ(n)}} is the Möbius function.

In the section Elementary properties, it was shown that if {{math|R(n)}} is the set of all {{mvar|n}}th roots of unity and {{math|P(n)}} is the set of primitive ones, {{math|R(n)}} is a disjoint union of the {{math|P(n)}}:

:$\backslash operatorname\{R\}(n)\; =\; \backslash bigcup\_\{d\; \backslash ,|\backslash ,\; n\}\backslash operatorname\{P\}(d),$

This implies

:$\backslash operatorname\{SR\}(n)\; =\; \backslash sum\_\{d\; \backslash ,|\backslash ,\; n\}\backslash operatorname\{SP\}(d).$

Applying the Möbius inversion formula gives

:$\backslash operatorname\{SP\}(n)\; =\; \backslash sum\_\{d\; \backslash ,|\backslash ,\; n\}\backslash mu(d)\backslash operatorname\{SR\}\backslash left(\backslash frac\{n\}\{d\}\backslash right).$

In this formula, if {{math|d < n}}, then {{math|1=SR({{sfrac|n|d}}) = 0}}, and for {{math|1=d = n}}: {{math|1=SR({{sfrac|n|d}}) = 1}}. Therefore, {{math|1=SP(n) = μ(n)}}.

This is the special case {{math|c_n(1)}} of Ramanujan's sum {{math|c_n(s)}},{{cite book

|last = Apostol |first = Tom M. |author-link = Tom M. Apostol

|year = 1976

|title = Introduction to Analytic Number Theory

|series = Undergraduate Texts in Mathematics |url = https://books.google.com/books?id=3yoBCAAAQBAJ&pg=PA160

|page = 160

|publisher = Springer

|doi = 10.1007/978-1-4757-5579-4|isbn = 978-1-4419-2805-4 }} defined as the sum of the {{mvar|s}}th powers of the primitive {{mvar|n}}th roots of unity:

:$c\_n(s)\; =\; \backslash sum\_\{a\; =\; 1\; \backslash atop\; \backslash gcd(a,\; n)\; =\; 1\}^n\; e^\{2\; \backslash pi\; i\; \backslash frac\{a\}\{n\}\; s\}.$

From the summation formula follows an orthogonality relationship: for {{math|1=j = 1, … , n}} and {{math|1=j′ = 1, … , n}}

:$\backslash sum\_\{k=1\}^\{n\}\; \backslash overline\{z^\{j\backslash cdot\; k\}\}\; \backslash cdot\; z^\{j\text{'}\backslash cdot\; k\}\; =\; n\; \backslash cdot\backslash delta\_\{j,j\text{'}\}$

where {{mvar|δ}} is the Kronecker delta and {{mvar|z}} is any primitive {{mvar|n}}th root of unity.

The {{math|n × n}} matrix {{mvar|U}} whose {{math|(j, k)}}th entry is

:$U\_\{j,k\}\; =\; n^\{-\backslash frac\{1\}\{2\}\}\backslash cdot\; z^\{j\backslash cdot\; k\}$

defines a discrete Fourier transform. Computing the inverse transformation using Gaussian elimination requires {{math|O(n³)}} operations. However, it follows from the orthogonality that {{mvar|U}} is unitary. That is,

:$\backslash sum\_\{k=1\}^\{n\}\; \backslash overline\{U\_\{j,k\}\}\; \backslash cdot\; U\_\{k,j\text{'}\}\; =\; \backslash delta\_\{j,j\text{'}\},$

and thus the inverse of {{mvar|U}} is simply the complex conjugate. (This fact was first noted by Gauss when solving the problem of trigonometric interpolation.) The straightforward application of {{mvar|U}} or its inverse to a given vector requires {{math|O(n²)}} operations. The fast Fourier transform algorithms reduces the number of operations further to {{math|O(n log n)}}.

==Cyclotomic polynomials==

{{Main|Cyclotomic polynomial}}

The zeros of the polynomial

:$p(z)\; =\; z^n\; -\; 1$

are precisely the {{mvar|n}}th roots of unity, each with multiplicity 1. The {{mvar|n}}th cyclotomic polynomial is defined by the fact that its zeros are precisely the primitive {{mvar|n}}th roots of unity, each with multiplicity 1.

: $\backslash Phi\_n(z)\; =\; \backslash prod\_\{k=1\}^\{\backslash varphi(n)\}(z-z\_k)$

where {{math|z₁, z₂, z₃, …, z_φ(n)}} are the primitive {{mvar|n}}th roots of unity, and {{math|φ(n)}} is Euler's totient function. The polynomial {{math|Φ_n(z)}} has integer coefficients and is an irreducible polynomial over the rational numbers (that is, it cannot be written as the product of two positive-degree polynomials with rational coefficients). The case of prime {{mvar|n}}, which is easier than the general assertion, follows by applying Eisenstein's criterion to the polynomial

:$\backslash frac\{(z+1)^n\; -\; 1\}\{(z+1)\; -\; 1\},$

and expanding via the binomial theorem.

Every {{mvar|n}}th root of unity is a primitive {{mvar|d}}th root of unity for exactly one positive divisor {{mvar|d}} of {{mvar|n}}. This implies that

:$z^n\; -\; 1\; =\; \backslash prod\_\{d\; \backslash ,|\backslash ,\; n\}\; \backslash Phi\_d(z).$

This formula represents the factorization of the polynomial {{math|zⁿ − 1}} into irreducible factors:

:$\backslash begin\{align\}$

z^1 -1 &= z-1 \\

z^2 -1 &= (z-1)(z+1) \\

z^3 -1 &= (z-1) (z^2 + z + 1) \\

z^4 -1 &= (z-1)(z+1) (z^2+1) \\

z^5 -1 &= (z-1) (z^4 + z^3 +z^2 + z + 1) \\

z^6 -1 &= (z-1)(z+1) (z^2 + z + 1) (z^2 - z + 1)\\

z^7 -1 &= (z-1) (z^6+ z^5 + z^4 + z^3 + z^2 + z + 1) \\

z^8 -1 &= (z-1)(z+1) (z^2+1) (z^4+1) \\

\end{align}

Applying Möbius inversion to the formula gives

:$\backslash Phi\_n(z)\; =\; \backslash prod\_\{d\; \backslash ,|\backslash ,\; n\}\backslash left(z^\backslash frac\{n\}\{d\}\; -\; 1\backslash right)^\{\backslash mu(d)\}\; =\; \backslash prod\_\{d\; \backslash ,|\backslash ,\; n\}\backslash left(z^d\; -\; 1\backslash right)^\{\backslash mu\backslash left(\backslash frac\{n\}\{d\}\backslash right)\},$

where {{math|μ}} is the Möbius function. So the first few cyclotomic polynomials are

:{{math|1=Φ₁(z) = z − 1}}

:{{math|1=Φ₂(z) = (z² − 1)⋅(z − 1)⁻¹ = z + 1}}

:{{math|1=Φ₃(z) = (z³ − 1)⋅(z − 1)⁻¹ = z² + z + 1}}

:{{math|1=Φ₄(z) = (z⁴ − 1)⋅(z² − 1)⁻¹ = z² + 1}}

:{{math|1=Φ₅(z) = (z⁵ − 1)⋅(z − 1)⁻¹ = z⁴ + z³ + z² + z + 1}}

:{{math|1=Φ₆(z) = (z⁶ − 1)⋅(z³ − 1)⁻¹⋅(z² − 1)⁻¹⋅(z − 1) = z² − z + 1}}

:{{math|1=Φ₇(z) = (z⁷ − 1)⋅(z − 1)⁻¹ = z⁶ + z⁵ + z⁴ + z³ + z² +z + 1}}

:{{math|1=Φ₈(z) = (z⁸ − 1)⋅(z⁴ − 1)⁻¹ = z⁴ + 1}}

If {{mvar|p}} is a prime number, then all the {{mvar|p}}th roots of unity except 1 are primitive {{mvar|p}}th roots. Therefore,

$$\backslash Phi\_p(z)\; =\; \backslash frac\{z^p\; -\; 1\}\{z\; -\; 1\}\; =\; \backslash sum\_\{k\; =\; 0\}^\{p\; -\; 1\}\; z^k.$$

Substituting any positive integer ≥ 2 for {{mvar|z}}, this sum becomes a radix repunit. Thus a necessary (but not sufficient) condition for a repunit to be prime is that its length be prime.

Note that, contrary to first appearances, not all coefficients of all cyclotomic polynomials are 0, 1, or −1. The first exception is {{math|Φ_{{num|105}}}}. It is not a surprise it takes this long to get an example, because the behavior of the coefficients depends not so much on {{mvar|n}} as on how many odd prime factors appear in {{mvar|n}}. More precisely, it can be shown that if {{mvar|n}} has 1 or 2 odd prime factors (for example, {{math|1=n = 150}}) then the {{mvar|n}}th cyclotomic polynomial only has coefficients 0, 1 or −1. Thus the first conceivable {{mvar|n}} for which there could be a coefficient besides 0, 1, or −1 is a product of the three smallest odd primes, and that is {{math|1=3 ⋅ 5 ⋅ 7 = 105}}. This by itself doesn't prove the 105th polynomial has another coefficient, but does show it is the first one which even has a chance of working (and then a computation of the coefficients shows it does). A theorem of Schur says that there are cyclotomic polynomials with coefficients arbitrarily large in absolute value. In particular, if $n\; =\; p\_1\; p\_2\; \backslash cdots\; p\_t,$ where are odd primes, $p\_1\; +p\_2>p\_t,$ and t is odd, then {{math|1 − t}} occurs as a coefficient in the {{mvar|n}}th cyclotomic polynomial.{{cite journal

|last = Lehmer|first = Emma|author-link = Emma Lehmer

|title = On the magnitude of the coefficients of the cyclotomic polynomial

|journal = Bulletin of the American Mathematical Society

|year = 1936

|volume = 42

|issue = 6

|pages = 389–392| doi=10.1090/S0002-9904-1936-06309-3 |doi-access = free

}}

Many restrictions are known about the values that cyclotomic polynomials can assume at integer values. For example, if {{mvar|p}} is prime, then {{math|d ∣ Φ_p(d)}} if and only if {{math|d ≡ 1 (mod p)}}.

Cyclotomic polynomials are solvable in radicals, as roots of unity are themselves radicals. Moreover, there exist more informative radical expressions for {{mvar|n}}th roots of unity with the additional property{{Cite journal

|last1 = Landau |first1 = Susan|authorlink1 = Susan Landau

|last2 = Miller |first2 = Gary L.|authorlink2 = Gary Miller (computer scientist)

|title = Solvability by radicals is in polynomial time

|journal = Journal of Computer and System Sciences

|volume = 30

|issue = 2

|pages = 179–208

|year = 1985

|doi = 10.1016/0022-0000(85)90013-3

|doi-access = }} that every value of the expression obtained by choosing values of the radicals (for example, signs of square roots) is a primitive {{mvar|n}}th root of unity. This was already shown by Gauss in 1797.{{Cite book|last=Gauss|first=Carl F.|author-link=Carl Friedrich Gauss|title=Disquisitiones Arithmeticae|pages=§§359–360|publisher=Yale University Press|year=1965|isbn=0-300-09473-6}} Efficient algorithms exist for calculating such expressions.{{cite web|last1=Weber|first1=Andreas|last2=Keckeisen|first2=Michael|title=Solving Cyclotomic Polynomials by Radical Expressions|url=http://cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/WeberKeckeisen99a.pdf|access-date=22 June 2007}}

The {{mvar|n}}th roots of unity form under multiplication a cyclic group of order {{mvar|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 {{mvar|n}}th root of unity.

The {{mvar|n}}th roots of unity form an irreducible representation of any cyclic group of order {{mvar|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.{{cite book

|last1 = Inui|first1 = Teturo

|last2 = Tanabe|first2 = Yukito

|last3 = Onodera|first3 = Yoshitaka

|title = Group Theory and Its Applications in Physics

|publisher = Springer

|year = 1996}}{{page needed|date=February 2023}} In particular, if a circulant Hermitian matrix is considered (for example, a discretized one-dimensional Laplacian with periodic boundaries{{cite journal

|last = Strang |first= Gilbert |author-link = Gilbert Strang

|title = The discrete cosine transform

|url = http://epubs.siam.org/sam-bin/dbq/article/33674

|journal = SIAM Review

|volume = 41

|issue = 1

|pages = 135–147

|year = 1999|doi= 10.1137/S0036144598336745 |bibcode= 1999SIAMR..41..135S }}), the orthogonality property immediately follows from the usual orthogonality of eigenvectors of Hermitian matrices.

{{Main|Cyclotomic field}}

By adjoining a primitive {{mvar|n}}th root of unity to $\backslash Q,$ one obtains the {{mvar|n}}th cyclotomic field $\backslash Q(\backslash exp(2\backslash pi\; i/n)).$This field contains all {{mvar|n}}th roots of unity and is the splitting field of the {{mvar|n}}th cyclotomic polynomial over $\backslash Q.$ The field extension $\backslash Q(\backslash exp(2\backslash pi\; i\; /n))/\backslash Q$ has degree φ(n) and its Galois group is naturally isomorphic to the multiplicative group of units of the ring $\backslash Z/n\backslash Z.$

As the Galois group of $\backslash Q(\backslash exp(2\backslash pi\; i\; /n))/\backslash Q$ is abelian, this is an abelian extension. Every subfield of a cyclotomic field is an abelian extension of the rationals. It follows that every nth root of unity may be expressed in term of k-roots, with various k not exceeding φ(n). In these cases Galois theory can be written out explicitly in terms of Gaussian periods: this theory from the Disquisitiones Arithmeticae of Gauss was published many years before Galois.The Disquisitiones was published in 1801, Galois was born in 1811, died in 1832, but wasn't published until 1846.

Conversely, every abelian extension of the rationals is such a subfield of a cyclotomic field – this is the content of a theorem of Kronecker, usually called the Kronecker–Weber theorem on the grounds that Weber completed the proof.

Image:Roots of unity, golden ratio.svg, the red points are the fifth roots of unity, and the black points are the sums of a fifth root of unity and its complex conjugate.]]

Image:Star polygon 8-2.svg

For {{math|1=n = 1, 2}}, both roots of unity {{num|1}} and {{num|−1}} are integers.

For three values of {{mvar|n}}, the roots of unity are quadratic integers:

• For {{math|1=n = 3, 6}} they are Eisenstein integers ({{math|1=D = −3}}).
• For {{math|1=n = 4}} they are Gaussian integers ({{math|1=D = −1}}): see Imaginary unit.

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

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

For {{math|1=n = 8}}, for any root of unity {{math|z + {{overline|z}}}} equals to either 0, ±2, or ±square root of 2 ({{math|1=D = 2}}).

For {{math|1=n = 12}}, for any root of unity, {{math|z + {{overline|z}}}} equals to either 0, ±1, ±2 or ±square root of 3 ({{math|1=D = 3}}).

• Argand system
• Circle group, the unit complex numbers
• Cyclotomic field
• Group scheme of roots of unity
• Dirichlet character
• Ramanujan's sum
• Witt vector
• Teichmüller character

{{Reflist}}

• {{Lang Algebra|edition=3r}}
• {{cite web|first=James S.|last=Milne|author-link=James Milne (mathematician)|title=Algebraic Number Theory|work=Course Notes|url=http://www.jmilne.org/math|year=1998}}
• {{cite web|first=James S.|last=Milne|title=Class Field Theory|work=Course Notes|url=http://www.jmilne.org/math|year=1997}}
• {{Neukirch ANT}}
• {{Cite book|first=Jürgen|last=Neukirch|title=Class Field Theory|publisher=Springer-Verlag|location=Berlin|year=1986|isbn=3-540-15251-2}}
• {{Cite book|first=Lawrence C.|last=Washington|author-link=Lawrence C. Washington|title=Introduction to Cyclotomic Fields|publisher=Springer-Verlag|location=New York|year=1997|edition=2nd|isbn=0-387-94762-0}}
• {{Cite book|author=Derbyshire, John|author-link=John Derbyshire|title=Unknown Quantity|chapter=Roots of Unity|year=2006|publisher=Joseph Henry Press|location=Washington, D.C.|isbn=0-309-09657-X|url=https://archive.org/details/isbn_9780309096577|url-access=registration}}

{{Algebraic numbers}}

{{DEFAULTSORT:Root of Unity}}

Category:Algebraic numbers

Category:Cyclotomic fields

Category:Polynomials

Category:1 (number)

Category:Complex numbers