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.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.

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 Ön) / 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.

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) ] )

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:=List(Irr(CharacterTable( "A5" )),ValuesOfClassFunction);
[ [ 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 ] ]

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