18 Cyclotomic Numbers

E returns the primitive n-th root of unity e_n = e^2πi/n. Cyclotomics are usually entered as sums of roots of unity, with rational coefficients, and irrational cyclotomics are displayed in the same way. (For special cyclotomics, see ATLAS irrationalities.)

gap> E(9); E(9)^3; E(6); E(12) / 3;
-E(9)^4-E(9)^7
E(3)
-E(3)^2
-1/3*E(12)^7

is the domain of all cyclotomics.

gap> E(9) in Cyclotomics; 37 in Cyclotomics; true in Cyclotomics;
true
true
false

As the cyclotomics are field elements the usual arithmetic operators +,-,* and / (and ^ to take powers by integers) are applicable. Note that ^ does not denote the conjugation of group elements, so it is not possible to explicitly construct groups of cyclotomics. (However, it is possible to compute the inverse and the multiplicative order of a nonzero cyclotomic.) Also, taking the k-th power of a root of unity z defines a Galois automorphism if and only if k is coprime to the conductor of z.

gap> E(5) + E(3); (E(5) + E(5)^4) ^ 2; E(5) / E(3); E(5) * E(3);
-E(15)^2-2*E(15)^8-E(15)^11-E(15)^13-E(15)^14
-2*E(5)-E(5)^2-E(5)^3-2*E(5)^4
E(15)^13
E(15)^8
gap> Order( E(5) ); Order( 1+E(5) );
5
infinity

Every object in the family CyclotomicsFamily lies in the category IsCyclotomic. This covers integers, rationals, proper cyclotomics, the object infinity (see Infinity), and unknowns (see Chapter Unknowns). All these objects except infinity and unknowns lie also in the category IsCyc, infinity lies in (and can be detected from) the category IsInfinity, and unknowns lie in IsUnknown.

gap> IsCyclotomic(0); IsCyclotomic(1/2*E(3)); IsCyclotomic( infinity );
true
true
true
gap> IsCyc(0); IsCyc(1/2*E(3)); IsCyc( infinity );
true
true
false

A cyclotomic is called integral or a cyclotomic integer if all coefficients of its minimal polynomial over the rationals are integers. Since the underlying basis of the external representation of cyclotomics is an integral basis (see Integral Bases of Abelian Number Fields), the subring of cyclotomic integers in a cyclotomic field is formed by those cyclotomics for which the external representation is a list of integers. For example, square roots of integers are cyclotomic integers (see ATLAS irrationalities), any root of unity is a cyclotomic integer, character values are always cyclotomic integers, but all rationals which are not integers are not cyclotomic integers.

gap> r:= ER( 5 );                # The square root of 5 is a cyclotomic integer.
E(5)-E(5)^2-E(5)^3+E(5)^4
gap> IsIntegralCyclotomic( r );  # It has integral coefficients.
true
gap> r2:= 1/2 * r;               # This is not a cyclotomic integer, ...
1/2*E(5)-1/2*E(5)^2-1/2*E(5)^3+1/2*E(5)^4
gap> IsIntegralCyclotomic( r2 );
false
gap> r3:= 1/2 * r - 1/2;         # ... but this is one.
E(5)+E(5)^4
gap> IsIntegralCyclotomic( r3 );
true

The operation Int can be used to find a cyclotomic integer near to an arbitrary cyclotomic. For rationals, Int returns the largest integer smaller or equal to the argument.

gap> Int( E(5)+1/2*E(5)^2 ); Int( 2/3*E(7)+3/2*E(4) );
E(5)
E(4)

The operation String returns for a cyclotomic a string corresponding to the way the cyclotomic is printed by ViewObj and PrintObj.

gap> String( E(5)+1/2*E(5)^2 ); String( 17/3 );
"E(5)+1/2*E(5)^2"
"17/3"

For an element cyc of a cyclotomic field, Conductor returns the smallest integer n such that cyc is contained in the n-th cyclotomic field. For a collection C of cyclotomics (for example a dense list of cyclotomics or a field of cyclotomics), Conductor returns the smallest integer n such that all elements of C are contained in the n-th cyclotomic field.

gap> Conductor( 0 ); Conductor( E(10) ); Conductor( E(12) );
1
5
12

returns the absolute value of a cyclotomic number cyc. At the moment only methods for rational numbers exist.

gap> AbsoluteValue(-3);
3

is a cyclotomic integer z (see IsIntegralCyclotomic) near to the cyclotomic cyc in the sense that the i-th coefficient in the external representation (see CoeffsCyc) of z is Int( c+1/2 ) where c is the i-th coefficient in the external representation of cyc. Expressed in terms of the Zumbroich basis (see Integral Bases of Abelian Number Fields), the coefficients of cyc w.r.t. this basis are rounded.

gap> RoundCyc( E(5)+1/2*E(5)^2 ); RoundCyc( 2/3*E(7)+3/2*E(4) );
E(5)+E(5)^2
-2*E(28)^3+E(28)^4-2*E(28)^11-2*E(28)^15-2*E(28)^19-2*E(28)^23-2*E(28)^27

Let cyc be a cyclotomic with conductor n. If N is not a multiple of n then CoeffsCyc returns fail because cyc cannot be expressed in terms of N-th roots of unity. Otherwise CoeffsCyc returns a list of length N with entry at position j equal to the coefficient of e^{2 πi (j−1)/N} if this root belongs to the N-th Zumbroich basis (see Integral Bases of Abelian Number Fields), and equal to zero otherwise. So we have cyc = CoeffsCyc(cyc,N) * List( [1..N], j -> E(N)^(j-1) ).

gap> cyc:= E(5)+E(5)^2;
E(5)+E(5)^2
gap> CoeffsCyc( cyc, 5 );  CoeffsCyc( cyc, 15 );  CoeffsCyc( cyc, 7 );
[ 0, 1, 1, 0, 0 ]
[ 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, -1, 0 ]
fail

For a cyclotomic number cyc (see IsCyclotomic), this function returns the smallest positive integer n such that n * cyc is a cyclotomic integer (see IsIntegralCyclotomic). For rational numbers cyc, the result is the same as that of DenominatorRat (see DenominatorRat).

gap> ExtRepOfObj( E(5) ); CoeffsCyc( E(5), 15 );
[ 0, 1, 0, 0, 0 ]
[ 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0 ]
gap> CoeffsCyc( 1+E(3), 9 ); CoeffsCyc( E(5), 7 );
[ 0, 0, 0, 0, 0, 0, -1, 0, 0 ]
fail

Given a cyclotomic root that is known to be a root of unity (this is not checked), DescriptionOfRootOfUnity returns a list [ n, e ] of coprime positive integers such that root = E(n)^e holds.

gap> E(9);  DescriptionOfRootOfUnity( E(9) );
-E(9)^4-E(9)^7
[ 9, 1 ]
gap> DescriptionOfRootOfUnity( -E(3) );
[ 6, 5 ]

IsGaussInt returns true if the object x is a Gaussian integer (see GaussianIntegers) and false otherwise. Gaussian integers are of the form a + b*E(4), where a and b are integers.

IsGaussRat returns true if the object x is a Gaussian rational (see GaussianRationals) and false otherwise. Gaussian rationals are of the form a + b*E(4), where a and b are rationals.

DefaultField (see DefaultField) for cyclotomics is defined to return the smallest cyclotomic field containing the given elements.

gap> Field( E(5)+E(5)^4 );  DefaultField( E(5)+E(5)^4 );
NF(5,[ 1, 4 ])
CF(5)

18.2 Infinity

infinity is a special GAP object that lies in CyclotomicsFamily. It is larger than all other objects in this family. infinity is mainly used as return value of operations such as Size and Dimension for infinite and infinite dimensional domains, respectively.

Note that no arithmetic operations are provided for infinity, in particular there is no problem to define what 0 * infinity or infinity - infinity means.

Often it is useful to distinguish infinity from ``proper'' cyclotomics. For that, infinity lies in the category IsInfinity but not in IsCyc, and the other cyclotomics lie in the category IsCyc but not in IsInfinity.

gap> s:= Size( Rationals );
infinity
gap> s = infinity; IsCyclotomic( s ); IsCyc( s ); IsInfinity( s );
true
true
false
true
gap> s in Rationals; s > 17;
false
true
gap> Set( [ s, 2, s, E(17), s, 19 ] );
[ 2, 19, E(17), infinity ]

18.3 Comparisons of Cyclotomics

To compare cyclotomics, the operators <, <=, =, >=, > and <> can be used, the result will be true if the first operand is smaller, smaller or equal, equal, larger or equal, larger, or unequal, respectively, and false otherwise.

Cyclotomics are ordered as follows: The relation between rationals is the natural one, rationals are smaller than irrational cyclotomics, and infinity is the largest cyclotomic. For two irrational cyclotomics with different conductors, the one with smaller conductor is regarded as smaller. Two irrational cyclotomics with same conductor are compared via their external representation.

For comparisons of cyclotomics and other GAP objects, see Section Comparisons.

gap> E(5) < E(6);      # the latter value has conductor 3
false
gap> E(3) < E(3)^2;    # both have conductor 3, compare the ext. repr.
false
gap> 3 < E(3); E(5) < E(7);
true
true

18.4 ATLAS Irrationalities

For N a positive integer, let z = E(N ) = exp(2 πi/N). The following so-called atomic irrationalities (see Chapter 7, Section 10 of CCN85) can be entered using functions. (Note that the values are not necessary irrational.)

EB(N ) = bN = 1 2 N−1 ∑ j=1  zj2, N ≡ 1 mod 2 EC(N ) = cN = 1 3 N−1 ∑ j=1  zj3, N ≡ 1 mod 3 ED(N ) = dN = 1 4 N−1 ∑ j=1  zj4, N ≡ 1 mod 4 EE(N ) = eN = 1 5 N−1 ∑ j=1  zj5, N ≡ 1 mod 5 EF(N ) = fN = 1 6 N−1 ∑ j=1  zj6, N ≡ 1 mod 6 EG(N ) = gN = 1 7 N−1 ∑ j=1  zj7, N ≡ 1 mod 7 EH(N ) = hN = 1 8 N−1 ∑ j=1  zj8, N ≡ 1 mod 8

(Note that in c_N, …, h_N, N must be a prime.)

For a rational number N, ER returns the square root √{N } of N, and EI returns √{−N }. By the chosen embedding of cyclotomic fields into the complex numbers, ER returns the positive square root if N is positive, and if N is negative then ER(N) = EI(-N). In any case, EI(N) = E(4) * ER(N).

ER is installed as method for the operation Sqrt (see Sqrt) for rational argument.

From a theorem of Gauss we know that

bN =        1 2 (−1+√N) if N ≡ 1 mod 4 1 2 (−1+i √N) if N ≡ −1 mod 4

So √N can be computed from b_N (see EB).

For given N, let n_k = n_k(N) be the first integer with multiplicative order exactly k modulo N, chosen in the order of preference

1, −1, 2, −2, 3, −3, 4, −4, ….

We define

EY(N ) = yn = z+zn (n=n2) EX(N ) = xn = z+zn+zn2 (n=n3) EW(N ) = wn = z+zn+zn2+zn3 (n=n4) EV(N ) = vn = z+zn+zn2+zn3+zn4 (n=n5) EU(N ) = un = z+zn+zn2+ …+zn5 (n=n6) ET(N ) = tn = z+zn+zn2+ …+zn6 (n=n7) ES(N ) = sn = z+zn+zn2+ …+zn7 (n=n8)

EM(N ) = mn = z−zn (n=n2) EL(N ) = ln = z−zn+zn2−zn3 (n=n4) EK(N ) = kn = z−zn+ …−zn5 (n=n6) EJ(N ) = jn = z−zn+ …−zn7 (n=n8)

Let n_k^(d) = n_k^(d)(N) be the d+1-th integer with multiplicative order exactly k modulo N, chosen in the order of preference defined above; we write n_k=n_k⁽⁰⁾,n_k^′=n_k⁽¹⁾, n_k^′′ = n_k⁽²⁾ and so on. These values can be computed as NK(N,k,d) = n_k^(d)(N); if there is no integer with the required multiplicative order, NK returns fail.

The algebraic numbers

yN′=yN(1),yN′′=yN(2),…, xN′,xN′′,…,jN′,jN′′,…

are obtained on replacing n_k in the above definitions by n_k^′,n_k^′′,…; they can be entered as

EY(N ,d ) = yN(d) EX(N ,d ) = xN(d) : EJ(N ,d ) = jn(d)

Let irratname be a string that describes an irrational value as described in Chapter 6, Section 10 of CCN85, that is, a linear combination of the atomic irrationalities introduced above. (The following definition is mainly copied from CCN85.) If q_N is such a value (e.g. y₂₄^′′) then linear combinations of algebraic conjugates of q_N are abbreviated as in the following examples:

2qN+3&5−4&7+&9 means 2 qN + 3 qN∗5 − 4 qN∗7 + qN∗9 4qN&3&5&7−3&4 means 4 (qN + qN∗3 + qN∗5 + qN∗7) − 3 qN∗11 4qN*3&5+&7 means 4 (qN∗3 + qN∗5) + qN∗7

To explain the ``ampersand'' syntax in general we remark that ``&k'' is interpreted as q_N^∗k, where q_N is the most recently named atomic irrationality, and that the scope of any premultiplying coefficient is broken by a + or − sign, but not by & or ∗k. The algebraic conjugations indicated by the ampersands apply directly to the atomic irrationality q_N, even when, as in the last example, q_N first appears with another conjugacy ∗k.

gap> EW(16,3); EW(17,2); ER(3); EI(3); EY(5); EB(9);
0
E(17)+E(17)^4+E(17)^13+E(17)^16
-E(12)^7+E(12)^11
E(3)-E(3)^2
E(5)+E(5)^4
1
gap> AtlasIrrationality( "b7*3" );
E(7)^3+E(7)^5+E(7)^6
gap> AtlasIrrationality( "y'''24" );
E(24)-E(24)^19
gap> AtlasIrrationality( "-3y'''24*13&5" );
3*E(8)-3*E(8)^3
gap> AtlasIrrationality( "3y'''24*13-2&5" );
-3*E(24)-2*E(24)^11+2*E(24)^17+3*E(24)^19
gap> AtlasIrrationality( "3y'''24*13-&5" );
-3*E(24)-E(24)^11+E(24)^17+3*E(24)^19
gap> AtlasIrrationality( "3y'''24*13-4&5&7" );
-7*E(24)-4*E(24)^11+4*E(24)^17+7*E(24)^19
gap> AtlasIrrationality( "3y'''24&7" );
6*E(24)-6*E(24)^19

18.5 Galois Conjugacy of Cyclotomics

For a cyclotomic cyc and an integer k, GaloisCyc returns the cyclotomic obtained by raising the roots of unity in the Zumbroich basis representation of cyc to the k-th power. If k is coprime to the integer n, GaloisCyc( ., k ) acts as a Galois automorphism of the n-th cyclotomic field (see Galois Groups of Abelian Number Fields); to get the Galois automorphisms themselves, use GaloisGroup (see GaloisGroup!of field).

The complex conjugate of cyc is GaloisCyc( cyc, -1 ), which can also be computed using ComplexConjugate (see ComplexConjugate).

For a list or matrix list of cyclotomics, GaloisCyc returns the list obtained by applying GaloisCyc to the entries of list.

For a cyclotomic number z, ComplexConjugate returns GaloisCyc( z, -1 ). For a quaternion z = c₁ e + c₂ i + c₃ j + c₄ k, ComplexConjugate returns c₁ e − c₂ i − c₃ j − c₄ k.

gap> GaloisCyc( E(5) + E(5)^4, 2 );
E(5)^2+E(5)^3
gap> GaloisCyc( E(5), -1 );           # the complex conjugate
E(5)^4
gap> GaloisCyc( E(5) + E(5)^4, -1 );  # this value is real
E(5)+E(5)^4
gap> GaloisCyc( E(15) + E(15)^4, 3 );
E(5)+E(5)^4
gap> ComplexConjugate( E(7) );
E(7)^6

If the cyclotomic cyc is an irrational element of a quadratic extension of the rationals then StarCyc returns the unique Galois conjugate of cyc that is different from cyc, otherwise fail is returned. In the first case, the return value is often called cyc ∗ (see Printing Character Tables).

gap> StarCyc( EB(5) ); StarCyc( E(5) );
E(5)^2+E(5)^3
fail

Let cyc be a cyclotomic integer that lies in a quadratic extension field of the rationals. Then we have cyc = (a + b √n) / d for integers a, b, n, d, such that d is either 1 or 2. In this case, Quadratic returns a record with the components a, b, root, d, ATLAS, and display; the values of the first four are a, b, n, and d, the ATLAS value is a (not necessarily shortest) representation of cyc in terms of the ATLAS irrationalities b_|n|, i_|n|, r_|n|, and the display value is a string that expresses cyc in GAP notation, corresponding to the value of the ATLAS component.

If cyc is not a cyclotomic integer or does not lie in a quadratic extension field of the rationals then fail is returned.

If the denominator d is 2 then necessarily n is congruent to 1 modulo 4, and r_n, i_n are not possible; we have cyc = x + y * EB( root ) with y = b, x = ( a + b ) / 2.

If d = 1, we have the possibilities i_|n| for n < −1, a + b * i for n = −1, a + b * r_n for n > 0. Furthermore if n is congruent to 1 modulo 4, also cyc = (a+b) + 2 * b * b_|n| is possible; the shortest string of these is taken as the value for the component ATLAS.

gap> Quadratic( EB(5) ); Quadratic( EB(27) );
rec( a := -1, b := 1, root := 5, d := 2, ATLAS := "b5", 
  display := "(-1+ER(5))/2" )
rec( a := -1, b := 3, root := -3, d := 2, ATLAS := "1+3b3", 
  display := "(-1+3*ER(-3))/2" )
gap> Quadratic(0); Quadratic( E(5) );
rec( a := 0, b := 0, root := 1, d := 1, ATLAS := "0", display := "0" )
fail

Let mat be a matrix of cyclotomics. GaloisMat calculates the complete orbits under the operation of the Galois group of the (irrational) entries of mat, and the permutations of rows corresponding to the generators of the Galois group.

If some rows of mat are identical, only the first one is considered for the permutations, and a warning will be printed.

GaloisMat returns a record with the components mat, galoisfams, and generators.

mat:

a list with initial segment being the rows of mat (not shallow copies of these rows); the list consists of full orbits under the action of the Galois group of the entries of mat defined above. The last rows in the list are those not contained in mat but must be added in order to complete the orbits; so if the orbits were already complete, mat and mat have identical rows.

galoisfams:

a list that has the same length as the mat component, its entries are either 1, 0, -1, or lists. galoisfams[i] = 1 means that mat[i] consists of rationals, i.e. [ mat[i] ] forms an orbit; galoisfams[i] = -1 means that mat[i] contains unknowns (see Chapter Unknowns); in this case [ mat[i] ] is regarded as an orbit, too, even if mat[i] contains irrational entries; if galoisfams[i] = [ l₁, l₂ ] is a list then mat[i] is the first element of its orbit in mat, l₁ is the list of positions of rows that form the orbit, and l₂ is the list of corresponding Galois automorphisms (as exponents, not as functions), so we have mat[ l₁[j] ][k] = GaloisCyc( mat[i][k], l₂[j] ); galoisfams[i] = 0 means that mat[i] is an element of a nontrivial orbit but not the first element of it.

generators:

a list of permutations generating the permutation group corresponding to the action of the Galois group on the rows of mat.

In the following example we temporarily increase the line length limit from its default value 80 to 84 in order to get a nicer output format.

gap> SizeScreen([ 84, ]);;
gap> GaloisMat( [ [ E(3), E(4) ] ] );
rec( 
  mat := [ [ E(3), E(4) ], [ E(3), -E(4) ], [ E(3)^2, E(4) ], [ E(3)^2, -E(4) ] ],
  galoisfams := [ [ [ 1, 2, 3, 4 ], [ 1, 7, 5, 11 ] ], 0, 0, 0 ], 
  generators := [ (1,2)(3,4), (1,3)(2,4) ] )
gap> SizeScreen([ 80, ]);;
gap> GaloisMat( [ [ 1, 1, 1 ], [ 1, E(3), E(3)^2 ] ] );
rec( mat := [ [ 1, 1, 1 ], [ 1, E(3), E(3)^2 ], [ 1, E(3)^2, E(3) ] ], 
  galoisfams := [ 1, [ [ 2, 3 ], [ 1, 2 ] ], 0 ], generators := [ (2,3) ] )

returns the list of rationalized rows of mat, which must be a matrix of cyclotomics. This is the set of sums over orbits under the action of the Galois group of the entries of mat (see GaloisMat), so the operation may be viewed as a kind of trace on the rows.

Note that no two rows of mat should be equal.

gap> mat:= [ [ 1, 1, 1 ], [ 1, E(3), E(3)^2 ], [ 1, E(3)^2, E(3) ] ];;
gap> RationalizedMat( mat );
[ [ 1, 1, 1 ], [ 2, -1, -1 ] ]

18.6 Internally Represented Cyclotomics

The implementation of an internally represented cyclotomic is based on a list of length equal to its conductor. This means that the internal representation of a cyclotomic does not refer to the smallest number field but the smallest cyclotomic field containing it. The reason for this is the wish to reflect the natural embedding of two cyclotomic fields into a larger one that contains both. With such embeddings, it is easy to construct the sum or the product of two arbitrary cyclotomics (in possibly different fields) as an element of a cyclotomic field.

The disadvantage of this approach is that the arithmetical operations are quite expensive, so the use of internally represented cyclotomics is not recommended for doing arithmetics over number fields, such as calculations with matrices of cyclotomics. But internally represented cyclotomics are good enough for dealing with irrationalities in character tables (see chapter Character Tables).

For the representation of cyclotomics one has to recall that the n-th cyclotomic field Q(e_n) is a vector space of dimension ϕ(n) over the rationals where ϕ denotes Euler's phi-function (see Phi).

A special integral basis of cyclotomic fields is chosen that allows one to easily convert arbitrary sums of roots of unity into the basis, as well as to convert a cyclotomic represented w.r.t. the basis into the smallest possible cyclotomic field. This basis is accessible in GAP, see Integral Bases of Abelian Number Fields for more information and references.

Note that the set of all n-th roots of unity is linearly dependent for n > 1, so multiplication is not the multiplication of the group ring Q〈e_n 〉; given a Q-basis of Q(e_n) the result of the multiplication (computed as multiplication of polynomials in e_n, using (e_n)ⁿ = 1) will be converted to the basis.

gap> E(5) * E(5)^2; ( E(5) + E(5)^4 ) * E(5)^2;
E(5)^3
E(5)+E(5)^3
gap> ( E(5) + E(5)^4 ) * E(5);
-E(5)-E(5)^3-E(5)^4

An internally represented cyclotomic is always represented in the smallest cyclotomic field it is contained in. The internal coefficients list coincides with the external representation returned by ExtRepOfObj.

Since the conductor of internally represented cyclotomics must be in the filter IsSmallIntRep, the biggest possible (though not very useful) conductor is 2²⁸−1. So the maximal cyclotomic field implemented in GAP is not really the field Q^ab.

gap> IsSmallIntRep( 2^28-1 );
true
gap> IsSmallIntRep( 2^28 );
false

It should be emphasized that one disadvantage of representing a cyclotomic in the smallest cyclotomic field (and not in the smallest field) is that arithmetic operations in a fixed small extension field of the rational number field are comparatively expensive. For example, take a prime integer p and suppose that we want to work with a matrix group over the field Q(√p). Then each matrix entry could be described by two rational coefficients, whereas the representation in the smallest cyclotomic field requires p−1 rational coefficients for each entry. So it is worth thinking about using elements in a field constructed with AlgebraicExtension (see AlgebraicExtension) when natural embeddings of cyclotomic fields are not needed.

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GAP 4 manual
March 2006