Root of unity explained

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 . 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 th roots of unity, except when is a multiple of the (positive) characteristic of the field.

General definition

An th root of unity, where is a positive integer, is a number satisfying the equation^{[1] [2]} $$z^n\; =\; 1.$$Unless otherwise specified, the roots of unity may be taken to be complex numbers (including the number 1, and the number −1 if is even, which are complex with a zero imaginary part), and in this case, the th roots of unity are^[3] $$\backslash exp\backslash left(\backslash frac\backslash right)=\backslash cos\backslash frac+i\backslash sin\backslash frac,\backslash qquad\; k=0,1,\backslash dots,\; n-1.$$

However, the defining equation of roots of unity is meaningful over any field (and even over any ring), 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^{[4] [5]}

z ^{n=1} and z ^{m} \ne 1 for m = 1 , 2 , 3 , \ldots, n -1 . If n is a prime number, then all th roots of unity, except 1, are primitive.^[6]

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,^[7] 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 th root of unity is its complex conjugate, and is also an th root of unity:^[8] (1)/(z) = z ^{-1} = z ^{n-1} = \bar z.

If is an th root of unity and then . Indeed, by the definition of congruence modulo n, for some integer, 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 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. (If where, then, which would imply that would not be primitive.) This implies that,, ..., , are all of the th roots of unity, since an th-degree polynomial equation over a field (in this case the field of complex numbers) has at most solutions.

From the preceding, it follows that, if is a primitive th root of unity, then

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

Let be a primitive th root of unity. A power of is a primitive th root of unity for

a = (n)/(\gcd(k,n)) , where \gcd(k,n) 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 a = (\operatorname{lcm)/(= (k,n)}{k} = (kn)/(k\gcd(k,n))) (n)/(\gcd(k,n)) .

Thus, if and are coprime, is also a primitive th root of unity, and therefore there are distinct primitive th roots of unity (where is Euler's totient function). 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 :

\operatorname{R}(n) = c up _{d} \operatorname{P}(d),

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

\sum _{d} \varphi(d) = n.

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

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

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

\omega \mapsto \omega ^{k}

induces an automorphism of

\Q(\omega) , which maps every th root of unity to its th power. Every automorphism of \Q(\omega) is obtained in this way, and these automorphisms form the Galois group of \Q(\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\mapsto \left(\omega \mapsto \omega ^{k\right)}

defines a group isomorphism between the units of the ring of integers modulo and the Galois group of

\Q(\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.

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

See main article: 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 \cos( 2 \pi/n). The roots of the minimal polynomial are just twice the real part; these roots form a cyclic Galois group.

Trigonometric expression

De Moivre's formula, which is valid for all real and integers, is

\left(\cos x + i \sin x\right) ^{n} = \cos nx + i \sin nx.

Setting gives a primitive th root of unity – one gets

\left(\cos (2 \pi)/(n) + i \sin (2 \pi)/(n) \right) ^{n} = \cos 2 \pi + i \sin 2 \pi = 1 , but\left(\cos (2 \pi)/(n) + i \sin (2 \pi)/(n) \right) ^{k} = \cos (2 k\pi)/(n) + i \sin (2 k\pi)/(n) ≠ 1 for . In other words, \cos (2 \pi)/(n) + i \sin (2 \pi)/(n) 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 (see the plots for and 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} = \cos x + i \sin x,

which is valid for all real, can be used to put the formula for the th roots of unity into the form

e ^{2 \pi i (k)/(n)} , 0 \le k < n.

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

\cos( 2 \pi k/n) , 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 (except for), the primitive th roots of unity are roots of an irreducible polynomial (over the integers) of lower degree, called the th cyclotomic polynomial, and often denoted . The degree of is given by Euler's totient function, which counts (among other things) the number of primitive th roots of unity.^[9] 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. (The trivial form

\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, there exists an expression built from integers by root extractions, additions, subtractions, multiplications, and divisions (and nothing else), 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 (possible values for a th root). (For more details see, below.)

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

r = z + (1 z)/() is twice the real part of . In other words, is a reciprocal polynomial, the polynomial R _{n} 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 R _{n} by solving the quadratic equation z ^{2-rz+1=0.} That is, the real part of the primitive root is (r 2 ,)/() and its imaginary part is \pm i\sqrt{ 1- \left( (r 2 \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 is a product of a power of two by a product (possibly empty) 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 (square) 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 (cube) roots of unity, which are the roots of this quadratic polynomial, are $$\backslash frac,\backslash \; \backslash frac\; .$$
• 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 $$\backslash frac4\; \backslash pm\; i\; \backslash frac,$$ where

\varepsilon may take the two values 1 and −1 (the same value in the two occurrences).

• As, there are two primitive sixth roots of unity, which are the negatives (and also the square roots) of the two primitive cube roots: $$\backslash frac,\backslash \; \backslash frac.$$
• 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 $$\backslash frac\backslash pm\; i\backslash sqrt,$$ 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 $$\backslash pm\backslash frac\; \backslash pm\; i\backslash frac.$$
• See Heptadecagon for the real part of a 17th root of unity.

Periodicity

If is a primitive th root of unity, then the sequence of powers

is -periodic (because for all values of), and the sequences of powers

for are all -periodic (because). Furthermore, the set of these sequences is a basis of the linear space of all -periodic sequences. This means that any -periodic sequence of complex numbers

can be expressed as a linear combination of powers of a primitive th root of unity:

x _{j} = \sum _{k} X _{k} ⋅ z ^{k} = X _{1} z ^{1 ⋅} + … + X _{n} ⋅ z ^{n} for some complex numbers and every integer .

This is a form of Fourier analysis. If is a (discrete) time variable, then is a frequency and is a complex amplitude.

Choosing for the primitive th root of unity

z = e ^{(2 \pi i)/(n)} = \cos (2 \pi)/(n) + i \sin (2 \pi)/(n) allows to be expressed as a linear combination of and :x _{j} = \sum _{k} A _{k} \cos (2 \pi jk)/(n) + \sum _{k} B _{k} \sin (2 \pi jk)/(n) . This is a discrete Fourier transform.

Summation

Let be the sum of all the th roots of unity, primitive or not. Then

\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 th roots of unity being the roots of the polynomial, their sum is the coefficient of degree, which is either 1 or 0 according whether or .

Alternatively, for there is nothing to prove, and for there exists a root – since the set of all the th roots of unity is a group,, so the sum satisfies, whence .

Let be the sum of all the primitive th roots of unity. Then

\operatorname{SP}(n) = \mu(n),

where is the Möbius function.

In the section Elementary properties, it was shown that if is the set of all th roots of unity and is the set of primitive ones, is a disjoint union of the :

\operatorname{R}(n) = c up _{d} \operatorname{P}(d),

This implies

\operatorname{SR}(n) = \sum _{d} \operatorname{SP}(d).

Applying the Möbius inversion formula gives

\operatorname{SP}(n) = \sum _{d} \mu(d)\operatorname{SR}\left( (n)/(d) \right).

In this formula, if, then, and for : . Therefore, .

This is the special case of Ramanujan's sum,^[10] defined as the sum of the th powers of the primitive th roots of unity:

c _{n(s)} = \sum ^{n} _{a =1 \atop \gcd(a, n) =1} e ^{2 \pi i (a)/(n) s} .

Orthogonality

From the summation formula follows an orthogonality relationship: for and

\sum ^{n} _{k =1} \overline{z ^{j ⋅} } \cdot z^ = n \cdot\delta_

where is the Kronecker delta and is any primitive th root of unity.

The matrix whose th entry is

U _{j,k} = n ^{- (1)/(2)} ⋅ z ^{j ⋅}

defines a discrete Fourier transform. Computing the inverse transformation using Gaussian elimination requires operations. However, it follows from the orthogonality that is unitary. That is,

\sum ^{n} _{k =1} \overline{U _{j,k} } \cdot U_ = \delta_,

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

Cyclotomic polynomials

See main article: Cyclotomic polynomial.

The zeros of the polynomial

p(z) = z ^{n} - 1 are precisely the th roots of unity, each with multiplicity 1. The th cyclotomic polynomial is defined by the fact that its zeros are precisely the primitive th roots of unity, each with multiplicity 1.\Phi _{n(z)} = \prod ^{\varphi(n)} _{k =1} (z - z _{k)} where are the primitive th roots of unity, and is Euler's totient function. The polynomial 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, which is easier than the general assertion, follows by applying Eisenstein's criterion to the polynomial((z +1 ) ^{n} -1)/((z +1 ) -1) ,

and expanding via the binomial theorem.

Every th root of unity is a primitive th root of unity for exactly one positive divisor of . This implies that

z ^{n} - 1 = \prod _{d} \Phi _{d(z).}

This formula represents the factorization of the polynomial into irreducible factors:

\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

\Phi _{n(z)} = \prod _{d} \left(z ^{(n)/(d)} - 1 \right) ^{\mu(d)} = \prod _{d} \left(z ^{d} - 1 \right) ^{\mu\left( (n)/(\right) d)} ,

where is the Möbius function. So the first few cyclotomic polynomials are

If is a prime number, then all the th roots of unity except 1 are primitive th roots. Therefore,$$\backslash Phi\_p(z)\; =\; \backslash frac\; =\; \backslash sum\_^\; z^k.$$Substituting any positive integer ≥&thinsp;2 for, this sum becomes a base 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 . It is not a surprise it takes this long to get an example, because the behavior of the coefficients depends not so much on as on how many odd prime factors appear in . More precisely, it can be shown that if has 1 or 2 odd prime factors (for example,) then the th cyclotomic polynomial only has coefficients 0, 1 or −1. Thus the first conceivable for which there could be a coefficient besides 0, 1, or −1 is a product of the three smallest odd primes, and that is . 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} … p _{t,} where p _{1} < p _{2} < … < p _{t} are odd primes, p _{1} + p _{2>p} _{t,} and t is odd, then occurs as a coefficient in the th cyclotomic polynomial.^[11]

Many restrictions are known about the values that cyclotomic polynomials can assume at integer values. For example, if is prime, then if and only if .

Cyclotomic polynomials are solvable in radicals, as roots of unity are themselves radicals. Moreover, there exist more informative radical expressions for th roots of unity with the additional property^[12] that every value of the expression obtained by choosing values of the radicals (for example, signs of square roots) is a primitive th root of unity. This was already shown by Gauss in 1797.^[13] Efficient algorithms exist for calculating such expressions.^[14]

Cyclic groups

The th roots of unity form under multiplication a cyclic group of order, 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 th root of unity.

The th roots of unity form an irreducible representation of any cyclic group of order . 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.

Cyclotomic fields

See main article: Cyclotomic field. By adjoining a primitive th root of unity to

\Q, one obtains the th cyclotomic field \Q(\exp( 2 \pi i/n)). This field contains all th roots of unity and is the splitting field of the th cyclotomic polynomial over \Q. The field extension \Q(\exp( 2 \pi i /n))/\Q has degree φ(n) and its Galois group is naturally isomorphic to the multiplicative group of units of the ring \Z/n\Z.

As the Galois group of

\Q(\exp( 2 \pi i /n))/\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.^[17]

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.

Relation to quadratic integers

For, both roots of unity and are integers.

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

• For they are Eisenstein integers .
• For they are Gaussian integers : see Imaginary unit.

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

For, none of the non-real roots of unity (which satisfy a quartic equation) is a quadratic integer, but the sum of each root with its complex conjugate (also a 5th root of unity) is an element of the ring . For two pairs of non-real 5th roots of unity these sums are inverse golden ratio and minus golden ratio.

For, for any root of unity equals to either 0, ±2, or ± .

For, for any root of unity, equals to either 0, ±1, ±2 or ± .

See also

• 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

References

• • Web site: James S.. Milne. James Milne (mathematician). Algebraic Number Theory. Course Notes. 1998.
• Web site: James S.. Milne. Class Field Theory. Course Notes. 1997.
  • Book: Neukirch, Jürgen. Class Field Theory. Springer-Verlag. Berlin. 1986. 3-540-15251-2.
• Book: Washington, Lawrence C.. Lawrence C. Washington. Introduction to Cyclotomic Fields. Springer-Verlag. New York. 1997. 2nd. 0-387-94762-0.
• Book: Derbyshire, John. John Derbyshire. Unknown Quantity. Roots of Unity. 2006. Joseph Henry Press. Washington, D.C.. 0-309-09657-X. registration.

Notes and References

1. Book: Hadlock, Charles R.. Charles Robert Hadlock. Field Theory and Its Classical Problems, Volume 14. Cambridge University Press. 2000. 978-0-88385-032-9. 84–86.
2. Book: Lang, Serge. Roots of unity. Algebra. Springer. 2002. 978-0-387-95385-4. 276–277. https://books.google.com/books?id=Fge-BwqhqIYC&pg=PA276.
3. Book: Meserve, Bruce E. . Fundamental Concepts of Algebra. 52. Dover Publications. 1982.
4. Book: Moskowitz, Martin A. . 2003. Adventure in Mathematics. World Scientific. 36. 9789812794949.
5. Book: Lidl . Rudolf. Pilz . Günter . Günter Pilz. 1984. Applied Abstract Algebra. Undergraduate Texts in Mathematics. 149. Springer. 10.1007/978-1-4615-6465-2. 978-0-387-96166-8.
6. Book: Morandi, Patrick . Field and Galois theory. Graduate Texts in Mathematics. 1996. 167. 74. Springer. 978-0-387-94753-2. 10.1007/978-1-4612-4040-2.
7. Book: Reilly, Norman R. . 2009. Introduction to Applied Algebraic Systems. 137. Oxford University Press. 978-0-19-536787-4.
8. Book: Rotman, Joseph J.. Joseph J. Rotman. Advanced Modern Algebra. 2015. 3rd. 1. 129. American Mathematical Society. 9781470415549 .
9. Book: Riesel, Hans . Hans Riesel. 1994. Prime Factorization and Computer Methods for Factorization. 306. Springer. 0-8176-3743-5.
10. Book: Apostol, Tom M. . Tom M. Apostol. 1976. Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics . 160. Springer. 10.1007/978-1-4757-5579-4. 978-1-4419-2805-4 .
11. Lehmer. Emma. Emma Lehmer. On the magnitude of the coefficients of the cyclotomic polynomial. Bulletin of the American Mathematical Society. 1936. 42. 6. 389–392. 10.1090/S0002-9904-1936-06309-3 . free.
12. Landau . Susan. Susan Landau. Miller . Gary L.. Gary Miller (computer scientist). Solvability by radicals is in polynomial time. Journal of Computer and System Sciences. 30. 2. 179–208. 1985. 10.1016/0022-0000(85)90013-3.
13. Book: Gauss, Carl F.. Carl Friedrich Gauss. Disquisitiones Arithmeticae. §§359–360. Yale University Press. 1965. 0-300-09473-6.
14. Web site: Weber. Andreas. Keckeisen. Michael. Solving Cyclotomic Polynomials by Radical Expressions. 22 June 2007.
15. Book: Inui. Teturo. Tanabe. Yukito. Onodera. Yoshitaka. Group Theory and Its Applications in Physics. Springer. 1996.
16. Strang . Gilbert . Gilbert Strang. The discrete cosine transform. SIAM Review. 41. 1. 135–147. 1999. 10.1137/S0036144598336745 . 1999SIAMR..41..135S .
17. The Disquisitiones was published in 1801, Galois was born in 1811, died in 1832, but wasn't published until 1846.