%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %A cyclotom.tex GAP documentation Thomas Breuer %% %A @(#)$Id: cyclotom.tex,v 3.11 1993/02/19 10:48:42 gap Rel $ %% %Y Copyright 1990-1992, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany %% %H $Log: cyclotom.tex,v $ %H Revision 3.11 1993/02/19 10:48:42 gap %H adjustments in line length and spelling %H %H Revision 3.10 1993/02/12 15:45:43 felsch %H more examples fixed %H %H Revision 3.9 1993/02/01 13:16:16 felsch %H examples fixed %H %H Revision 3.8 1992/04/07 23:05:55 martin %H changed the author line %H %H Revision 3.7 1992/03/27 13:56:54 sam %H removed "'" in index entries %H %H Revision 3.6 1992/03/11 15:48:05 sam %H renamed chapter "Number Fields" to "Subfields of Cyclotomic Fields" %H %H Revision 3.5 1992/02/13 15:37:26 sam %H renamed 'Automorphisms' to 'GaloisGroup' %H %H Revision 3.4 1992/01/14 14:37:27 martin %H changed an overlong line %H %H Revision 3.3 1992/01/14 14:01:12 sam %H adjusted citations %H %H Revision 3.2 1991/12/30 09:37:48 martin %H changed two section titles slightly %H %H Revision 3.1 1991/12/30 08:07:00 sam %H initial revision under RCS %H %% \Chapter{Cyclotomics}\index{type!cyclotomic}\index{irrationalities} {\GAP} allows computations in abelian extension fields of the rational field $Q$, i.e., fields with abelian Galois group over $Q$. These fields are described in chapter "Subfields of Cyclotomic Fields". They are subfields of *cyclotomic fields* $Q_n = Q(e_n)$ where $e_n = e^{\frac{2\pi i}{n}}$ is a primitive $n$--th root of unity. Their elements are called *cyclotomics*. The internal representation of a cyclotomic does not refer to the smallest number field but the smallest cyclotomic field containing it (the so--called *conductor*). This is because it is easy to embed two cyclotomic fields in a larger one that contains both, i.e., there is a natural way to get the sum or the product of two arbitrary cyclotomics as element of a cyclotomic field. The disadvantage is that the arithmetical operations are too expensive to do arithmetics in number fields, e.g., calculations in a matrix ring over a number field. But it suffices to deal with irrationalities in character tables (see "Character Tables"). (And in fact, the comfortability of working with the natural embeddings is used there in many situations which did not actually afford it \ldots) All functions that take a field extension as ---possibly optional--- argument, e.g., 'Trace' or 'Coefficients' (see chapter "Fields"), are described in chapter "Subfields of Cyclotomic Fields". This chapter informs about:\\ the representation of cyclotomics in {\GAP} (see "More about Cyclotomics"),\\ access to the internal data (see "NofCyc", "CoeffsCyc")\\ integral elements of number fields (see "Cyclotomic Integers", "IntCyc", "RoundCyc"),\\ characteristic functions (see "IsCyc", "IsCycInt"),\\ comparison and arithmetical operations of cyclotomics (see "Comparisons of Cyclotomics", "Operations for Cyclotomics"),\\ functions concerning Galois conjugacy of cyclotomics (see "GaloisCyc", "StarCyc"), or lists of them (see "GaloisMat", "RationalizedMat"),\\ some special cyclotomics, as defined in \cite{CCN85} (see "ATLAS irrationalities", "Quadratic") The external functions are in the file 'LIBNAME/\"cyclotom.g\"'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{More about Cyclotomics}\index{cyclotomic field elements} \index{E}\index{roots of unity} Elements of number fields (see chapter "Subfields of Cyclotomic Fields"), cyclotomics for short, are arithmetical objects like rationals and finite field elements; they are not implemented as records ---like groups--- or e.g. with respect to a character table (although character tables may be the main interest for cyclotomic arithmetics). 'E( )' returns the primitive -th root of unity $e_n = e^{\frac{2\pi i}{n}}$. Cyclotomics are usually entered as (and irrational cyclotomics are always displayed as) sums of roots of unity with rational coefficients. (For special cyclotomics, see "ATLAS irrationalities".) | gap> E(9); E(9)^3; E(6); E(12) / 3; -E(9)^4-E(9)^7 # the root needs not to be an element of the base E(3) -E(3)^2 -1/3*E(12)^7| For the representation of cyclotomics one has to recall that the cyclotomic field $Q_n = Q(e_n)$ is a vector space of dimension $\varphi(n)$ over the rationals where $\varphi$ denotes Euler\'s phi-function (see "Phi"). 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 $Q\langle e_n \rangle$; given a $Q$-basis of $Q_n$ the result of the multiplication (computed as multiplication of polynomials in $e_n$, using $(e_n)^n = 1$) will be converted to the base. | 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| Cyclotomics are always represented in the smallest cyclotomic field they are contained in. Together with the choice of a fixed base this means that two cyclotomics are equal if and only if they are equally represented. Addition and multiplication of two cyclotomics represented in $Q_n$ and $Q_m$, respectively, is computed in the smallest cyclotomic field containing both\:\ $Q_{'Lcm'(n,m)}$. Conversely, if the result is contained in a smaller cyclotomic field the representation is reduced to the minimal such field. The base, the base conversion and the reduction to the minimal cyclotomic field are described in~\cite{Zum89}, more about the base can be found in "ZumbroichBase". Since $n$ must be a 'short integer', the maximal cyclotomic field implemented in {\GAP} is not really the field $Q^{ab}$. The biggest allowed (though not very useful) $n$ is 65535. There is a global variable 'Cyclotomics' in {\GAP}, a record that stands for the domain of all cyclotomics (see chapter "Subfields of Cyclotomic Fields"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Cyclotomic Integers} A cyclotomic is called *integral* or *cyclotomic integer* if all coefficients of its minimal polynomial are integers. Since the base used is an integral base (see "ZumbroichBase"), the subring of cyclotomic integers in a cyclotomic field is formed by those cyclotomics which have not only rational but integral coefficients in their representation as sums of roots of unity. 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. (See "IsCycInt") | gap> ER( 5 ); # The square root of 5 is a cyclotomic E(5)-E(5)^2-E(5)^3+E(5)^4 # integer, it has integral coefficients. gap> 1/2 * ER( 5 ); # This is not a cyclotomic integer, \ldots 1/2*E(5)-1/2*E(5)^2-1/2*E(5)^3+1/2*E(5)^4 gap> 1/2 * ER( 5 ) - 1/2; # \ldots but this is one. E(5)+E(5)^4| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{IntCyc} 'IntCyc( )' returns the cyclotomic integer (see "Cyclotomic Integers") with Zumbroich base coefficients (see "ZumbroichBase") 'List( , x -> Int( x ) )' where is the vector of Zumbroich base coefficients of the cyclotomic ; see also "RoundCyc". | gap> IntCyc( E(5)+1/2*E(5)^2 ); IntCyc( 2/3*E(7)+3/2*E(4) ); E(5) E(4)| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{RoundCyc} 'RoundCyc( )' returns the cyclotomic integer (see "Cyclotomic Integers") with Zumbroich base coefficients (see "ZumbroichBase") 'List( , x -> Int( x+1/2 ) )' where is the vector of Zumbroich base coefficients of the cyclotomic ; see also "IntCyc". | 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| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{IsCyc}\index{test!for a cyclotomic} 'IsCyc( )' returns 'true' if is a cyclotomic, and 'false' otherwise. Will signal an error if is an unbound variable. | gap> IsCyc( 0 ); IsCyc( E(3) ); IsCyc( 1/2 * E(3) ); IsCyc( IsCyc ); true true true false| 'IsCyc' is an internal function. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{IsCycInt}\index{test!for a cyclotomic integer} 'IsCycInt( )' returns 'true' if is a cyclotomic integer (see "Cyclotomic Integers"), 'false' otherwise. Will signal an error if is an unbound variable. | gap> IsCycInt( 0 ); IsCycInt( E(3) ); IsCycInt( 1/2 * E(3) ); true true false| 'IsCycInt' is an internal function. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{NofCyc} 'NofCyc( )'\\ 'NofCyc( )' returns the smallest positive integer $n$ for which the cyclotomic is resp.\ for which all cyclotomics in the list are contained in $Q_n = Q( e^{\frac{2 \pi i}{n}} ) = Q( 'E()' )$. | gap> NofCyc( 0 ); NofCyc( E(10) ); NofCyc( E(12) ); 1 5 12| 'NofCyc' is an internal function. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{CoeffsCyc}\index{coefficients!for cyclotomics} 'CoeffsCyc( , )' If is a cyclotomic which is contained in $Q_n$, 'CoeffsCyc( , )' returns a list of length where the entry at position is the coefficient of $'E()'^{i-1}$ in the internal representation of as element of the cyclotomic field $Q_n$ (see "More about Cyclotomics", "ZumbroichBase")\:\ $ = [1] + [2]\ 'E()'^1 + \ldots + [n]\ 'E()'^{n-1}$. *Note* that all positions which do not belong to base elements of $Q_n$ contain zeroes. | gap> CoeffsCyc( E(5), 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 ] Error, no representation of in 7th roots of unity| 'CoeffsCyc' calls the internal function 'COEFFSCYC'\: 'COEFFSCYC( )' is equivalent to 'CoeffsCyc( , NofCyc( ) )', see "NofCyc". %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Comparisons of Cyclotomics}\index{operators!for 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 inequal, respectively, and 'false' otherwise. Cyclotomics are ordered as follows\:\ The relation between rationals is as usual, and rationals are smaller than irrational cyclotomics. For two irrational cyclotomics , which lie in different minimal cyclotomic fields, we have $ \< $ if and only if $'NofCyc()' \< 'NofCyc()'$); if $'NofCyc()' = 'NofCyc()'$), that one is smaller that has the smaller coefficient vector, i.e., $ \leq $ if and only if $'COEFFSCYC()' \leq 'COEFFSCYC()'$. You can compare cyclotomics with objects of other types; all objects which are not cyclotomics are larger than cyclotomics. | gap> E(5) < E(6); # the latter value lies in $Q_3$ false gap> E(3) < E(3)^2; # both lie in $Q_3$, so compare coefficients false gap> 3 < E(3); E(5) < E(7); true true gap> E(728) < (1,2); true| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Operations for Cyclotomics} \index{operators!for cyclotomics} The operators '+', '-', '\*', '/' are used for addition, subtraction, multiplication and division of two cyclotomics; note that division by 0 causes an error. '+' and '-' can also be used as unary operators; '\^' is used for exponentiation of a cyclotomic with an integer; this is in general *not* equal to Galois conjugation. | 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| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{GaloisCyc}\index{galois conjugation}\index{galois automorphism} 'GaloisCyc( , )' returns the cyclotomic obtained on raising the roots of unity in the representation of the cyclotomic to the -th power. If is represented in the field $Q_n$ and is a fixed integer relative prime to , 'GaloisCyc( ., )' acts as a Galois automorphism of $Q_n$ (see "GaloisGroup for Number Fields"); to get Galois automorphisms as functions, use "GaloisGroup" 'GaloisGroup'. | 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| 'GaloisCyc' is an internal function. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{ATLAS irrationalities}\index{$b_N$}\index{$c_N$}\index{$d_N$} \index{$e_N$}\index{$f_N$}\index{$g_N$}\index{$h_N$}\index{$i_N$} \index{$j_N$}\index{$k_N$}\index{$l_N$}\index{$m_N$}\index{$r_N$} \index{$s_N$}\index{$t_N$}\index{$u_N$}\index{$v_N$}\index{$w_N$} \index{$x_N$}\index{$y_N$}\index{$n_k$}\index{atomic irrationalities} \index{EB}\index{EC}\index{ED}\index{EE}\index{EF}\index{EG}\index{EH} \index{EI}\index{EJ}\index{EK}\index{EL}\index{EM}\index{ER}\index{ES} \index{ET}\index{EU}\index{EV}\index{EW}\index{EX}\index{EY}\index{NK} 'EB( )', 'EC( )', \ldots, 'EH( )',\\ 'EI( )', 'ER( )',\\ 'EJ( )', 'EK( )', 'EL( )', 'EM( )',\\ 'EJ( , )', 'EK( , )', 'EL( , )', 'EM( , )',\\ 'ES( )', 'ET( )', \ldots, 'EY( )',\\ 'ES( , )', 'ET( , )', \ldots, 'EY( , )',\\ 'NK( , , )' For $N$ a positive integer, let $z = 'E()' = e^{2 \pi i / N}$. The following so-called atomic irrationalities (see~\cite[Chapter 7, Section 10]{CCN85}) can be entered by functions (Note that the values are not necessary irrational.)\: \[\begin{array}{llllll} 'EB()' & = & b_N & = & \frac{1}{2}\sum_{j=1}^{N-1}z^{j^{2}} & (N\equiv 1\bmod 2)\\ 'EC()' & = & c_N & = & \frac{1}{3}\sum_{j=1}^{N-1}z^{j^{3}} & (N\equiv 1\bmod 3)\\ 'ED()' & = & d_N & = & \frac{1}{4}\sum_{j=1}^{N-1}z^{j^{4}} & (N\equiv 1\bmod 4)\\ 'EE()' & = & e_N & = & \frac{1}{5}\sum_{j=1}^{N-1}z^{j^{5}} & (N\equiv 1\bmod 5)\\ 'EF()' & = & f_N & = & \frac{1}{6}\sum_{j=1}^{N-1}z^{j^{6}} & (N\equiv 1\bmod 6)\\ 'EG()' & = & g_N & = & \frac{1}{7}\sum_{j=1}^{N-1}z^{j^{7}} & (N\equiv 1\bmod 7)\\ 'EH()' & = & h_N & = & \frac{1}{8}\sum_{j=1}^{N-1}z^{j^{8}} & (N\equiv 1\bmod 8) \end{array}\] (Note that in $c_N, \ldots, h_N$, must be a prime.) \[\begin{array}{lllll} 'ER()' & = & \sqrt{N}\\ 'EI()' & = & i \sqrt{N} & = & \sqrt{-N}\\ \end{array}\] From a theorem of Gauss we know that \[ b_N = \left\{ \begin{array}{llll} \frac{1}{2}(-1+\sqrt{N}) & {\rm if} & N\equiv 1 & \bmod 4 \\ \frac{1}{2}(-1+i\sqrt{N}) & {\rm if} & N\equiv -1 & \bmod 4 \end{array}\right. ,\] so $\sqrt{N}$ can be (and in fact is) computed from $b_N$. For given , let $n_k = n_k(N)$ be the first integer with multiplicative order exactly modulo , chosen in the order of preference \[ 1, -1, 2, -2, 3, -3, 4, -4, \ldots\ .\] We have \[\begin{array}{llllll} 'EY()' & = & y_n & = & z+z^n &(n = n_2)\\ 'EX()' & = & x_n & = & z+z^n+z^{n^2} &(n=n_3)\\ 'EW()' & = & w_n & = & z+z^n+z^{n^2}+z^{n^3} &(n=n_4)\\ 'EV()' & = & v_n & = & z+z^n+z^{n^2}+z^{n^3}+z^{n^4} &(n=n_5)\\ 'EU()' & = & u_n & = & z+z^n+z^{n^2}+\ldots +z^{n^5} &(n=n_6)\\ 'ET()' & = & t_n & = & z+z^n+z^{n^2}+\ldots +z^{n^6} &(n=n_7)\\ 'ES()' & = & s_n & = & z+z^n+z^{n^2}+\ldots +z^{n^7} &(n=n_8) \end{array}\] \[\begin{array}{llllll} 'EM()' & = & m_n & = & z-z^n &(n=n_2)\\ 'EL()' & = & l_n & = & z-z^n+z^{n^2}-z^{n^3} &(n=n_4)\\ 'EK()' & = & k_n & = & z-z^n+\ldots -z^{n^5} &(n=n_6)\\ 'EJ()' & = & j_n & = & z-z^n+\ldots -z^{n^7} &(n=n_8) \end{array}\] Let $n_k^{(d)} = n_k^{(d)}(N)$ be the $d+1$-th integer with multiplicative order exactly modulo , chosen in the order of preference defined above; we write $n_k=n_k^{(0)},n_k^{\prime}=n_k^{(1)}, n_k^{\prime\prime} = n_k^{(2)}$ and so on. These values can be computed as 'NK(,,)'$ = n_k^{(d)}(N)$; if there is no integer with the required multiplicative order, 'NK' will return 'false'. The algebraic numbers \[y_N^{\prime}=y_N^{(1)},y_N^{\prime\prime}=y_N^{(2)},\ldots, x_N^{\prime},x_N^{\prime\prime},\ldots, j_N^{\prime},j_N^{\prime\prime},\ldots\] are obtained on replacing $n_k$ in the above definitions by $n_k^{\prime},n_k^{\prime\prime},\ldots$; they can be entered as \[\begin{array}{lll} 'EY(,)' & = & y_N^{(d)}\\ 'EX(,)' & = & x_N^{(d)}\\ & \vdots \\ 'EJ(,)' & = & j_n^{(d)} \end{array}\] | gap> EW(16,3); EW(17,2); ER(3); ER(-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| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{StarCyc}\index{galois conjugate!unique} 'StarCyc( )' If is an irrational element of a quadratic number field (i.e. if is a quadratic irrationality), 'StarCyc( )' returns the unique Galois conjugate of that is different from ; this is often called $\ast$ (see "DisplayCharTable"). Otherwise 'false' is returned. | gap> StarCyc( EB(5) ); StarCyc( E(5) ); E(5)^2+E(5)^3 false| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Quadratic}\index{quadratic irrationalities} \index{quadratic number fields} 'Quadratic( )' If is a cyclotomic integer that is contained in a quadratic number field over the rationals, it can be written as $ = \frac{ a + b \sqrt{n} }{d}$ with integers $a$, $b$, $n$ and $d$, where $d$ is either 1 or 2. In this case 'Quadratic( )' returns a record with fields 'a', 'b', 'root', 'd' and 'ATLAS' where the first four mean the integers mentioned above, and the last one is a string that is a (not necessarily shortest) representation of by $b_m$, $i_m$ or $r_m$ for $m = '\|root\|'$ (see "ATLAS irrationalities"). If is not a quadratic irrationality or not a cyclotomic integer, 'false' is returned. | gap> Quadratic( EB(5) ); Quadratic( EB(27) ); rec( a := -1, b := 1, root := 5, d := 2, ATLAS := "b5" ) rec( a := -1, b := 3, root := -3, d := 2, ATLAS := "1+3b3" ) gap> Quadratic(0); Quadratic( E(5) ); rec( a := 0, b := 0, root := 1, d := 1, ATLAS := "0" ) false| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{GaloisMat}\index{galois conjugate characters} 'GaloisMat( )' must be a matrix of cyclotomics (or possibly unknowns, see "Unknown"). The conjugate of a row in under a particular Galois automorphism is defined pointwise. If consists of full orbits under this action then the Galois group of its entries acts on as a permutation group, otherwise the orbits must be completed before. 'GaloisMat( )' returns a record with fields 'mat', 'galoisfams' and 'generators'\: 'mat':\\ a list with initial segment (*not* a copy of ); the list consists of full orbits under the action of the Galois group of the entries of defined above. The last entries are those rows which had to be added to complete the orbits; so if they were already complete, and 'mat' have identical entries. 'galoisfams':\\ a list that has the same length as 'mat'; 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; in this case $[ 'mat[i]' ]$ is regarded as an orbit, too, even if 'mat[i]' contains irrational entries.\\ If $'galoisfams[i]' = [ l_1, l_2 ]$ is a list then 'mat[i]' is the first element of its orbit in 'mat'; $l_1$ is the list of positions of rows which form the orbit, and $l_2$ is the list of corresponding Galois automorphisms (as exponents, not as functions); so we have $'mat'[ l_1[j] ][k] = 'GaloisCyc'( 'mat'[i][k], l_2[j] )$.\\ $'galoisfams[i]' = 0$ 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'. Note that should be a set, i.e. no two rows should be equal. Otherwise only the first row of some equal rows is considered for the permutations, and a warning is printed. | 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> 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) ] )| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{RationalizedMat}\index{rational characters} 'RationalizedMat( )' returns the set of rationalized rows of , i.e. the set of sums over orbits under the action of the Galois group of the elements of (see "GaloisMat"). This may be viewed as a kind of trace operation for the rows. Note that should be a set, i.e. no two rows should be equal. | gap> mat:= CharTable( "A5" ).irreducibles; [ [ 1, 1, 1, 1, 1 ], [ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ], [ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ], [ 4, 0, 1, -1, -1 ], [ 5, 1, -1, 0, 0 ] ] gap> RationalizedMat( mat ); [ [ 1, 1, 1, 1, 1 ], [ 6, -2, 0, 1, 1 ], [ 4, 0, 1, -1, -1 ], [ 5, 1, -1, 0, 0 ] ]|