  
  [1X6 [33X[0;0YRelatively free Algebras[133X[101X
  
  [33X[0;0YAs  described  in [Eic11], the nilpotent quotient algorithm can also be used
  to  determine  certain  relatively free algebras; that is, algebras that are
  free within a variety.[133X
  
  
  [1X6.1 [33X[0;0YComputing Kurosh Algebras[133X[101X
  
  [1X6.1-1 KuroshAlgebra[101X
  
  [33X[1;0Y[29X[2XKuroshAlgebra[102X( [3Xd[103X, [3Xn[103X, [3XF[103X ) [32X function[133X
  
  [33X[0;0Ydetermines  a  nilpotent  table  for  the  largest  associative algebra on [22Xd[122X
  generators over the field [22XF[122X so that every element [22Xa[122X of the algebra satisfies
  [22Xa^n = 0[122X.[133X
  
  [1X6.1-2 ExpandExponentLaw[101X
  
  [33X[1;0Y[29X[2XExpandExponentLaw[102X( [3XT[103X, [3Xn[103X ) [32X function[133X
  
  [33X[0;0Ysuppose  that  [22XT[122X  is  the  nilpotent table of a Kurosh algebra of exponent [22Xn[122X
  defined  over a prime field. This function determines polynomials describing
  the   corresponding   Kurosh   algebras   over  all  fields  with  the  same
  characteristic as the prime field.[133X
  
  
  [1X6.2 [33X[0;0YA Library of Kurosh Algebras[133X[101X
  
  [33X[0;0YThe  package  contains a library of Kurosh algebras. This can be accessed as
  follows.[133X
  
  [1X6.2-1 KuroshAlgebraByLib[101X
  
  [33X[1;0Y[29X[2XKuroshAlgebraByLib[102X( [3Xd[103X, [3Xn[103X, [3XF[103X ) [32X function[133X
  
  [33X[0;0YAt present, the library contains the Kurosh algebras for [22Xn=2[122X, [22X(d,n) = (2,3)[122X,
  [22X(d,n)  =  (3,3)[122X and [22XF = ℚ[122X or [22X|F| ∈ {2,3,4}[122X, [22X(d,n) = (4,3)[122X and [22XF = ℚ[122X or [22X|F| ∈
  {2,3,4}[122X, [22X(d,n) = (2,4)[122X and [22XF = ℚ[122X or [22X|F| ∈ {2,3,4,9}[122X, [22X(d,n) = (2,5)[122X and [22XF = ℚ[122X
  or [22X|F| ∈ {2,3,4,5,8,9}[122X.[133X
  
  
  [1X6.3 [33X[0;0YExample of accessing the library of Kurosh algebras[133X[101X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XKuroshAlgebra(2,2,Rationals);[127X[104X
    [4X[28X... some printout ..[128X[104X
    [4X[28Xrec( bas := [ [ 1, 0, 0, 0 ], [ 0, 1, 1, 0 ], [ 0, 0, 0, 1 ], [ 0, 1, 0, 0 ] ][128X[104X
    [4X[28X    , com := false, dim := 3, fld := Rationals, rnk := 2, [128X[104X
    [4X[28X  tab := [ [ [ 0, 0, 0 ], [ 0, 0, -1 ], [ 0, 0, 0 ] ], [128X[104X
    [4X[28X      [ [ 0, 0, 1 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ] ], wds := [ ,, [ 2, 1 ] ], [128X[104X
    [4X[28X  wgs := [ 1, 1, 2 ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y [133X
  
