This file is indexed.

/usr/share/gap/lib/gaussian.gi is in gap-libs 4r6p5-3.

This file is owned by root:root, with mode 0o644.

The actual contents of the file can be viewed below.

  1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
#############################################################################
##
#W  gaussian.gi                 GAP library                  Martin Schönert
##
##
#Y  Copyright (C)  1997,  Lehrstuhl D für Mathematik,  RWTH Aachen,  Germany
#Y  (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland
#Y  Copyright (C) 2002 The GAP Group
##
##  This file contains methods for Gaussian rationals and Gaussian integers.
##
##  Gaussian rationals are elements of the form $a + b * I$ where $I$ is the
##  square root of -1 and $a,b$ are rationals.
##  Note that $I$ is written as `E(4)', i.e., as a fourth root of unity in
##  {\GAP}.
##  Gauss was the first to investigate such numbers, and already proved that
##  the ring of integers of this field, i.e., the elements of the form
##  $a + b * I$ where $a,b$ are integers, forms a Euclidean Ring.
##  It follows that this ring is a Unique Factorization Domain.
##


#############################################################################
##
#M  \in( <n>, GaussianIntegers )  . . . membership test for Gaussian integers
##
##  Gaussian integers are of the form `<a> + <b> * E(4)', where <a> and <b>
##  are integers.
##
InstallMethod( \in,
    "for Gaussian integers",
    IsElmsColls,
    [ IsCyc, IsGaussianIntegers ],
    function( cyc, GaussianIntegers )
    return IsCycInt( cyc ) and 4 mod Conductor( cyc ) = 0;
    end );


#############################################################################
##
#M  Basis( GaussianIntegers ) . . . . . . . . . . . . . for Gaussian integers
##
InstallMethod( Basis,
    "for Gaussian integers (delegate to `CanonicalBasis')",
    [ IsGaussianIntegers ], CANONICAL_BASIS_FLAGS,
    CanonicalBasis );

    
#############################################################################
##
#M  CanonicalBasis( GaussianIntegers )  . . . . . . . . for Gaussian integers
##
DeclareRepresentation(
    "IsCanonicalBasisGaussianIntegersRep", IsAttributeStoringRep,
    [ "conductor", "zumbroichbase" ] );

InstallMethod( CanonicalBasis,
    "for Gaussian integers",
    [ IsGaussianIntegers ],
    function( GaussianIntegers )
    local B;

    B:= Objectify( NewType( FamilyObj( GaussianIntegers ),
                                IsFiniteBasisDefault
                            and IsCanonicalBasis
                            and IsCanonicalBasisGaussianIntegersRep ),
#T generalize this to integral rings of cyclotomics!
                   rec() );
    SetUnderlyingLeftModule( B, GaussianIntegers );
    SetIsIntegralBasis( B, true );
    SetBasisVectors( B, Immutable( [ 1, E(4) ] ) );
    B!.conductor:= 4;
    B!.zumbroichbase := [ 0, 1 ];

    # Return the basis.
    return B;
    end );

    
#############################################################################
##
#M  Coefficients( <B>, <z> )  . for the canon. basis of the Gaussian integers
##
InstallMethod( Coefficients,
    "for canon. basis of Gaussian integers, and cyclotomic",
    IsCollsElms,
    [ IsBasis and IsCanonicalBasis and IsCanonicalBasisGaussianIntegersRep,
      IsCyc ],
    function( B, z )
    local N,
          coeffs,
          F;

    F:= UnderlyingLeftModule( B );
    if not z in F then return fail; fi;

    N:= B!.conductor;

    # Get the Zumbroich basis representation of <z> in `N'-th roots.
    coeffs:= CoeffsCyc( z, N );
    if coeffs = fail then return fail; fi;

    # Get the Zumbroich basis coefficients (basis $B_{n,1}$)
    coeffs:= coeffs{ B!.zumbroichbase + 1 };

    # Return the list of coefficients.
    return coeffs;
    end );


#############################################################################
##
#M  Quotient( GaussianIntegers, <n>, <m> )
##
InstallMethod( Quotient,
    "for Gaussian integers",
    IsCollsElmsElms,
    [ IsGaussianIntegers, IsCyc, IsCyc ],
    function ( GaussianIntegers, x, y )
    local   q;
    q := x / y;
    if not IsCycInt( q )  then
        q := fail;
    fi;
    return q;
    end );


#############################################################################
##
#M  StandardAssociateUnit( GaussianIntegers, <x> )  . . for Gaussian integers
##
##  The standard associate of <x> is an associated element <y> of <x> that
##  lies in the  first quadrant of the complex plane.
##  That is <y> is that element from `<x> * [1,-1,E(4),-E(4)]' that has
##  positive real part and nonnegative imaginary part.
##  (This is the generalization of `Abs' (see "Abs") for Gaussian integers.)
##
##  This function returns the unit <z> equal to <y> / <x>. The default
##  StandardAssociate method then uses this to compute the standard associate.
##
InstallMethod( StandardAssociateUnit,
    "for Gaussian integers",
    IsCollsElms,
    [ IsGaussianIntegers, IsCyc ],
    function ( GaussianIntegers, x )
    if not IsGaussInt( x )  then
        Error( "<x> must lie in <GaussianIntegers>" );
    elif IsRat(x)  and 0 <= x  then
        return 1;
    elif IsRat(x)  then
        return -1;
    elif 0 <  COEFFS_CYC(x)[1]       and 0 <= COEFFS_CYC(x)[2]       then
        return 1;
    elif      COEFFS_CYC(x)[1] <= 0  and 0 <  COEFFS_CYC(x)[2]       then
        return -E(4);
    elif      COEFFS_CYC(x)[1] <  0  and      COEFFS_CYC(x)[2] <= 0  then
        return -1;
    else
        return E(4);
    fi;
    end );


#############################################################################
##
#M  EuclideanDegree( GaussianIntegers, <n> )
##
InstallMethod( EuclideanDegree,
    "for Gaussian integers",
    IsCollsElms,
    [ IsGaussianIntegers, IsCyc ],
    function( GaussianIntegers, x )
    if IsGaussInt( x ) then
      return x * GaloisCyc( x, -1 );
    else
      Error( "<x> must lie in <GaussianIntegers>" );
    fi;
    end );


#############################################################################
##
#M  EuclideanRemainder( GaussianIntegers, <n>, <m> )
##
InstallMethod( EuclideanRemainder,
    "for Gaussian integers",
    IsCollsElmsElms,
    [ IsGaussianIntegers, IsCyc, IsCyc ],
    function ( GaussianIntegers, x, y )
    if IsGaussInt( x ) and IsGaussInt( y ) then
      return x - RoundCyc( x/y ) * y;
    else
      Error( "<x> and <y> must lie in <GaussianIntegers>" );
    fi;
    end );


#############################################################################
##
#M  EuclideanQuotient( GaussianIntegers, <x>, <y> )
##
InstallMethod( EuclideanQuotient,
    "for Gaussian integers",
    IsCollsElmsElms,
    [ IsGaussianIntegers, IsCyc, IsCyc ],
    function ( GaussianIntegers, x, y )
    if IsGaussInt( x ) and IsGaussInt( y ) then
      return RoundCyc( x/y );
    else
      Error( "<x> and <y> must lie in <GaussianIntegers>" );
    fi;
    end );


#############################################################################
##
#M  QuotientRemainder( GaussianIntegers, <x>, <y> )
##
InstallMethod( QuotientRemainder,
    "for Gaussian integers",
    IsCollsElmsElms,
    [ IsGaussianIntegers, IsCyc, IsCyc ],
    function ( GaussianIntegers, x, y )
    local   q;
    if IsGaussInt( x ) and IsGaussInt( y ) then
      q := RoundCyc(x/y);
      return [ q, x-q*y ];
    else
      Error( "<x> and <y> must lie in <GaussianIntegers>" );
    fi;
    end );


#############################################################################
##
#M  IsPrime( GaussianIntegers, <n> )
##
InstallMethod( IsPrime,
    "for Gaussian integers and integer",
    IsCollsElms,
    [ IsGaussianIntegers, IsInt ],
    function ( GaussianIntegers, x )
    return x mod 4 = 3  and IsPrimeInt( x );
    end );

InstallMethod( IsPrime,
    "for Gaussian integers and cyclotomic",
    IsCollsElms,
    [ IsGaussianIntegers, IsCyc ],
    function ( GaussianIntegers, x )
    if IsGaussInt( x ) then
      return IsPrimeInt( x * GaloisCyc( x, -1 ) );
    else
      Error( "<x> must lie in <GaussianIntegers>" );
    fi;
    end );


#############################################################################
##
#M  Factors( GaussianIntegers, <x> )
##
InstallMethod( Factors,
    "for Gaussian integers",
    IsCollsElms,
    [ IsGaussianIntegers, IsCyc ],
    function ( GaussianIntegers, x )
    local   facs,       # factors (result)
            prm,        # prime factors of the norm
            tsq;        # representation of `prm' as $x^2 + y^2$

    # handle trivial cases
    if x in [ 0, 1, -1, E(4), -E(4) ]  then
        return [ x ];
    elif not IsGaussInt( x ) then
        Error( "<x> must lie in <GaussianIntegers>" );
    fi;

    # loop over all factors of the norm of x
    facs := [];
    for prm in Set( FactorsInt( EuclideanDegree( GaussianIntegers, x ) ) ) do

        # $p = 2$ and primes $p = 1$ mod 4 split according to $p = x^2 + y^2$
        if prm = 2  or prm mod 4 = 1  then
            tsq := TwoSquares( prm );
            while IsCycInt( x / (tsq[1]+tsq[2]*E(4)) )  do
                Add( facs, (tsq[1]+tsq[2]*E(4)) );
                x := x / (tsq[1]+tsq[2]*E(4));
            od;
            while IsCycInt( x / (tsq[2]+tsq[1]*E(4)) )  do
                Add( facs, (tsq[2]+tsq[1]*E(4)) );
                x := x / (tsq[2]+tsq[1]*E(4));
            od;

        # primes $p = 3$ mod 4 stay prime
        else
            while IsCycInt( x / prm )  do
                Add( facs, prm );
                x := x / prm;
            od;
        fi;

    od;

    Assert( 1, x in [ 1, -1, E(4), -E(4) ],
            "'Factors' for Gaussian integers: Cofactor must be a unit\n" );

    # the first factor takes the unit
    facs[1] := x * facs[1];

    # return the result
    return facs;
    end );


#T #############################################################################
#T ##
#T #F  GaussianRationalsOps.CharPol(<GaussRat>,<x>)  . .  characteristic polynom
#T #F                                                     of a Gaussian rational
#T ##
#T GaussianRationalsOps.CharPol := function ( GaussRat, x )
#T     return [ x * GaloisCyc(x,-1), -x-GaloisCyc(x,-1), 1 ];
#T end;
#T 
#T 
#T #############################################################################
#T ##
#T #F  GaussianRationalsOps.MinPol(<GaussRat>,<x>) . . . . . . . minimal polynom
#T #F                                                     of a Gaussian rational
#T ##
#T GaussianRationalsOps.MinPol := function ( GaussRat, x )
#T     if IsRat( x )  then
#T         return [ -x, 1 ];
#T     else
#T         return [ x * GaloisCyc(x,-1), -x-GaloisCyc(x,-1), 1 ];
#T     fi;
#T end;


#############################################################################
##
#E