  
  [1X4 Constructing forms and basic functionality[0X
  
  In  this  chapter,  all  operations  to construct sesquilinear and quadratic
  forms are listed, along with their basic attributes and properties.
  
  
  [1X4.1 Important filters[0X
  
  
  [1X4.1-1 Categories for forms[0X
  
  [2X> IsBilinearForm____________________________________________________[0XCategory
  [2X> IsHermitianForm___________________________________________________[0XCategory
  [2X> IsSesquilinearForm________________________________________________[0XCategory
  [2X> IsQuadraticForm___________________________________________________[0XCategory
  [2X> IsForm____________________________________________________________[0XCategory
  [2X> IsForm____________________________________________________________[0XCategory
  [2X> IsTrivialForm_____________________________________________________[0XCategory
  
  The   categories  [10XIsBilinearForm[0m  and  [10XIsHermitianForm[0m  are  categories  for
  bilinear  and  hermitian forms, respectively. They are disjoint and are both
  contained in the category [10XIsSesquilinearForm[0m.
  
  Quadratic   forms   are  contained  in  the  category  [10XIsQuadraticForm[0m.  The
  categories  [10XIsSesquilinearForm[0m and [10XIsQuadraticForm[0m are disjoint and are both
  contained in the category [10XIsForm[0m.
  
  The  user  is  allowed to construct the trivial form (mapping all vectors to
  the  zero  element  of  the  field).  The  trivial  form is an object in the
  category  [10XIsTrivialForm[0m.  This  category is contained in [10XIsForm[0m and disjoint
  from [10XIsSesquilinearForm[0m and [10XIsQuadraticForm[0m.
  
  
  [1X4.1-2 Representation for forms[0X
  
  [2X> IsFormRep___________________________________________________[0XRepresentation
  
  Every  form  is  represented  by  a  matrix,  the  base  field  and a string
  describing the ``type'' of the form.
  
  
  [1X4.2 Constructing forms using a matrix[0X
  
  [1X4.2-1 BilinearFormByMatrix[0m
  
  [2X> BilinearFormByMatrix( [0X[3Xmatrix, field[0X[2X ) ___________________________[0Xoperation
  [2X> BilinearFormByMatrix( [0X[3Xmatrix[0X[2X ) __________________________________[0Xoperation
  [6XReturns:[0X  a bilinear form
  
  The  argument  [3Xmatrix[0m  must be a symmetric, or skew-symmetric, square matrix
  over the finite field [3Xfield[0m. The argument [3Xfield[0m is an optional argument, and
  if  it  is not given, then we assume that the [13Xdefining field[0m of the bilinear
  form  is  the smallest field containing the entries of matrix. Below we give
  an  example  where  the  defining  field  can  make  a  difference  in  some
  applications.  As it is only possible to construct reflexive bilinear forms,
  it  is  checked whether the matrix [3Xmatrix[0m is symmetric or skew symmetric. If
  matrix  [3Xmatrix[0m is not symmetric nor skew symmetric, then an error message is
  returned.  The output is a bilinear form (i.e., an object in [10XIsBilinearForm[0m)
  with  Gram  matrix  [3Xmatrix[0m  and  defining  field [3Xfield[0m. (See [14X3.1[0m for more on
  bilinear forms).
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> mat := IdentityMat(4, GF(9));[0X
    [4X[ [ Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ], [0X
    [4X  [ 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3) ], [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0 ] ][0X
    [4Xgap> form := BilinearFormByMatrix(mat,GF(9));[0X
    [4X< bilinear form >[0X
    [4Xgap> Display(form);[0X
    [4XBilinear form[0X
    [4XGram Matrix:[0X
    [4X 1 . . .[0X
    [4X . 1 . .[0X
    [4X . . 1 .[0X
    [4X . . . 1[0X
    [4Xgap> mat := [[0*Z(2),Z(16)^12,0*Z(2),Z(4)^2,Z(16)^13],[0X
    [4X>    [Z(16)^12,0*Z(2),0*Z(2),Z(16)^11,Z(16)],[0X
    [4X>    [0*Z(2),0*Z(2),0*Z(2),Z(4)^2,Z(16)^3],[0X
    [4X>    [Z(4)^2,Z(16)^11,Z(4)^2,0*Z(2),Z(16)^3],[0X
    [4X>    [Z(16)^13,Z(16),Z(16)^3,Z(16)^3,0*Z(2) ]];[0X
    [4X[ [ 0*Z(2), Z(2^4)^12, 0*Z(2), Z(2^2)^2, Z(2^4)^13 ], [0X
    [4X  [ Z(2^4)^12, 0*Z(2), 0*Z(2), Z(2^4)^11, Z(2^4) ], [0X
    [4X  [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2^2)^2, Z(2^4)^3 ], [0X
    [4X  [ Z(2^2)^2, Z(2^4)^11, Z(2^2)^2, 0*Z(2), Z(2^4)^3 ], [0X
    [4X  [ Z(2^4)^13, Z(2^4), Z(2^4)^3, Z(2^4)^3, 0*Z(2) ] ][0X
    [4Xgap> form := BilinearFormByMatrix(mat,GF(16));[0X
    [4X< bilinear form >[0X
    [4Xgap> Display(form);[0X
    [4XBilinear form[0X
    [4XGram Matrix:[0X
    [4Xz = Z(16)[0X
    [4X    . z^12    . z^10 z^13[0X
    [4X z^12    .    . z^11  z^1[0X
    [4X    .    .    . z^10  z^3[0X
    [4X z^10 z^11 z^10    .  z^3[0X
    [4X z^13  z^1  z^3  z^3    .[0X
    [4Xgap> mat := [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]*Z(7)^0;[0X
    [4X[ [ Z(7)^0, 0*Z(7), 0*Z(7), 0*Z(7) ], [ 0*Z(7), Z(7)^0, 0*Z(7), 0*Z(7) ], [0X
    [4X  [ 0*Z(7), 0*Z(7), 0*Z(7), Z(7)^0 ], [ 0*Z(7), 0*Z(7), Z(7)^0, 0*Z(7) ] ][0X
    [4Xgap> form := BilinearFormByMatrix(mat);[0X
    [4X< bilinear form >[0X
    [4Xgap> WittIndex(form);[0X
    [4X1[0X
    [4Xgap> form := BilinearFormByMatrix(mat,GF(49));[0X
    [4X< bilinear form >[0X
    [4Xgap> WittIndex(form);[0X
    [4X2[0X
    [4X [0X
  [4X------------------------------------------------------------------[0X
  
  [1X4.2-2 QuadraticFormByMatrix[0m
  
  [2X> QuadraticFormByMatrix( [0X[3Xmatrix, field[0X[2X ) __________________________[0Xoperation
  [2X> QuadraticFormByMatrix( [0X[3Xmatrix[0X[2X ) _________________________________[0Xoperation
  [6XReturns:[0X  a quadratic form
  
  The argument [3Xmatrix[0m must be a square matrix over the finite field [3Xfield[0m. The
  argument  [3Xfield[0m  is  an  optional  argument, and if it is not given, then we
  assume  that  the  [13Xdefining field[0m of the bilinear form is the smallest field
  containing  the  entries  of  matrix.  Below  we  give  an example where the
  defining field can make a difference in some applications. Any square matrix
  determines a quadratic form, but the Gram matrix is recomputed so that it is
  an upper triangle matrix. The output is a quadratic form (i.e., an object in
  [10XIsQuadraticForm[0m)  with  defining  field [3Xfield[0m. (See [14X3.2[0m for more on bilinear
  forms).
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> mat := [[1,0,0,0],[0,3,0,0],[0,0,0,6],[0,0,6,0]]*Z(7)^0;[0X
    [4X[ [ Z(7)^0, 0*Z(7), 0*Z(7), 0*Z(7) ], [ 0*Z(7), Z(7), 0*Z(7), 0*Z(7) ], [0X
    [4X  [ 0*Z(7), 0*Z(7), 0*Z(7), Z(7)^3 ], [ 0*Z(7), 0*Z(7), Z(7)^3, 0*Z(7) ] ][0X
    [4Xgap> form := QuadraticFormByMatrix(mat,GF(7));[0X
    [4X< quadratic form >[0X
    [4Xgap> Display(form);[0X
    [4XQuadratic form[0X
    [4XGram Matrix:[0X
    [4X 1 . . .[0X
    [4X . 3 . .[0X
    [4X . . . 5[0X
    [4X . . . .[0X
    [4Xgap> gf := GF(2^2);[0X
    [4XGF(2^2)[0X
    [4Xgap> mat := InvariantQuadraticForm( SO(-1, 4, 4) )!.matrix;[0X
    [4X[ [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [0X
    [4X  [ 0*Z(2), 0*Z(2), Z(2^2)^2, Z(2)^0 ], [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2^2)^2 ] ][0X
    [4Xgap> form := QuadraticFormByMatrix( mat, gf );[0X
    [4X< quadratic form >[0X
    [4Xgap> Display(form);[0X
    [4XQuadratic form[0X
    [4XGram Matrix:[0X
    [4Xz = Z(4)[0X
    [4X   .   1   .   .[0X
    [4X   .   .   .   .[0X
    [4X   .   . z^2   1[0X
    [4X   .   .   . z^2[0X
    [4X [0X
  [4X------------------------------------------------------------------[0X
  
  The  following  example  shows how using the argument [3Xfield[0m has influence on
  the properties of the constructed form.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> mat := [0X
    [4X> [[Z(2)^0,Z(2)^0,0*Z(2),0*Z(2)],[0*Z(2),Z(2)^0,0*Z(2),0*Z(2)], [0X
    [4X>  [0*Z(2),0*Z(2),0*Z(2),Z(2)^0],[0*Z(2),0*Z(2),0*Z(2),0*Z(2)]];[0X
    [4X[ [ Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], [0X
    [4X  [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ][0X
    [4Xgap> form := QuadraticFormByMatrix(mat);[0X
    [4X< quadratic form >[0X
    [4Xgap> WittIndex(form);[0X
    [4X1[0X
    [4Xgap> form := QuadraticFormByMatrix(mat,GF(4));[0X
    [4X< quadratic form >[0X
    [4Xgap> WittIndex(form);[0X
    [4X2[0X
    [4X [0X
  [4X------------------------------------------------------------------[0X
  
  [1X4.2-3 HermitianFormByMatrix[0m
  
  [2X> HermitianFormByMatrix( [0X[3Xmatrix, field[0X[2X ) __________________________[0Xoperation
  [6XReturns:[0X  a quadratic form
  
  The  argument [3Xmatrix[0m must be a hermitian square matrix over the finite field
  [3Xfield[0m, and [3Xfield[0m has square order. The field must be specified, since we can
  only determine the smallest field containing the entries of [3Xmatrix[0m. As it is
  only  possible  to  construct  reflexive  sesquilinear  forms, it is checked
  whether  the  matrix  is a hermitian matrix, and if not, an error message is
  returned.  The  output  is a hermitian sesquilinear form (i.e., an object in
  [10XIsHermitianForm[0m)  with Gram matrix [3Xmatrix[0m and defining field [3Xfield[0m. (See [14X3.1[0m
  for more on hermitian forms).
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> gf := GF(3^2);[0X
    [4XGF(3^2)[0X
    [4Xgap> mat := IdentityMat(4, gf);[0X
    [4X[ [ Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ], [0X
    [4X  [ 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3) ], [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0 ] ][0X
    [4Xgap> form := HermitianFormByMatrix( mat, gf );[0X
    [4X< hermitian form >[0X
    [4Xgap> Display(form);[0X
    [4XHermitian form[0X
    [4XGram Matrix:[0X
    [4X 1 . . .[0X
    [4X . 1 . .[0X
    [4X . . 1 .[0X
    [4X . . . 1[0X
    [4Xgap> mat := [[Z(11)^0,0*Z(11),0*Z(11)],[0*Z(11),0*Z(11),Z(11)],[0X
    [4X>     [0*Z(11),Z(11),0*Z(11)]];[0X
    [4X[ [ Z(11)^0, 0*Z(11), 0*Z(11) ], [ 0*Z(11), 0*Z(11), Z(11) ], [0X
    [4X  [ 0*Z(11), Z(11), 0*Z(11) ] ][0X
    [4Xgap> form := HermitianFormByMatrix(mat,GF(121));[0X
    [4X< hermitian form >[0X
    [4Xgap> Display(form);[0X
    [4XHermitian form[0X
    [4XGram Matrix:[0X
    [4X  1  .  .[0X
    [4X  .  .  2[0X
    [4X  .  2  .[0X
    [4X [0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X4.3 Constructing forms using a polynomial[0X
  
  Suppose  that  f  is  a  sesquilinear  form on an n-dimensional vectorspace.
  Consider  a  vector  x  with coordinates x_1,...,x_n with x_i indeterminates
  over  the  field. Then f(x,x) is a polynomial in n indeterminates. When f is
  alternating,  f(x,x)  is  identically  zero,  but in all other cases, f(x,x)
  determines f completely.
  
  Conversely,  suppose  that  Q  is  a  quadratic  form  on  an  n-dimensional
  vectorspace.  Consider  a  vector  x  with  coordinates x_1,...,x_n with x_i
  indeterminates   over   the   field.   Then   Q(x)  is  a  polynomial  in  n
  indeterminates, and Q(x) determines Q completely.
  
  [5XForms[0m  provides functionality to construct bilinear, hermitian and quadratic
  forms using an appropriate polynomial.
  
  [1X4.3-1 BilinearFormByPolynomial[0m
  
  [2X> BilinearFormByPolynomial( [0X[3Xpoly, r, n[0X[2X ) __________________________[0Xoperation
  [2X> BilinearFormByPolynomial( [0X[3Xpoly, r[0X[2X ) _____________________________[0Xoperation
  [6XReturns:[0X  a bilinear form
  
  The  argument  [3Xpoly[0m  must  be  a  polynomial  in  the polynomial ring [3Xr[0m. The
  (optional) last argument is the dimension for the underlying vector space of
  the  resulting  form,  which  by  default  is  the  number of indeterminates
  specified  by  [3Xpoly[0m.  It  is checked whether the polynomial is a homogeneous
  polynomial  of degree two over the given field, and if not, an error message
  is returned. It is not possible to construct a nontrivial bilinear form from
  a  polynomial  in even characteristic. The output is a bilinear (orthogonal)
  form in the category [10XIsBilinearForm[0m. (See [14X3.1[0m for more on bilinear forms).
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> r := PolynomialRing( GF(11), 4);[0X
    [4XGF(11)[x_1,x_2,x_3,x_4][0X
    [4Xgap> vars := IndeterminatesOfPolynomialRing( r );[0X
    [4X[ x_1, x_2, x_3, x_4 ][0X
    [4Xgap> pol := vars[1]*vars[2]+vars[3]*vars[4];[0X
    [4Xx_1*x_2+x_3*x_4[0X
    [4Xgap> form := BilinearFormByPolynomial(pol, r, 4);[0X
    [4X< bilinear form >[0X
    [4Xgap> Display(form);[0X
    [4XBilinear form[0X
    [4XGram Matrix:[0X
    [4X  .  6  .  .[0X
    [4X  6  .  .  .[0X
    [4X  .  .  .  6[0X
    [4X  .  .  6  .[0X
    [4XPolynomial: x_1*x_2+x_3*x_4[0X
    [4Xgap> r := PolynomialRing(GF(4),2);[0X
    [4XGF(2^2)[x_1,x_2][0X
    [4Xgap> pol := r.1*r.2;[0X
    [4Xx_1*x_2[0X
    [4Xgap> form := BilinearFormByPolynomial(pol,r);[0X
    [4XError, No orthogonal form can be associated with a quadratic polynomial in eve[0X
    [4Xn characteristic[0X
    [4X called from[0X
    [4XBilinearFormByPolynomial( pol, pring, n ) called from[0X
    [4X<function>( <arguments> ) called from read-eval-loop[0X
    [4XEntering break read-eval-print loop ...[0X
    [4Xyou can 'quit;' to quit to outer loop, or[0X
    [4Xyou can 'return;' to continue[0X
    [4Xbrk> quit;[0X
    [4X [0X
  [4X------------------------------------------------------------------[0X
  
  [1X4.3-2 QuadraticFormByPolynomial[0m
  
  [2X> QuadraticFormByPolynomial( [0X[3Xpoly, r, n[0X[2X ) _________________________[0Xoperation
  [2X> QuadraticFormByPolynomial( [0X[3Xpoly, r[0X[2X ) ____________________________[0Xoperation
  [6XReturns:[0X  a quadratic form
  
  The  argument  [3Xpoly[0m  must  be  a  polynomial  in  the polynomial ring [3Xr[0m. The
  (optional) last argument is the dimension for the underlying vector space of
  the  resulting  form,  which  by  default  is  the  number of indeterminates
  specified  by  [3Xpoly[0m.  It  is checked whether the polynomial is a homogeneous
  polynomial  of degree two over the given field, and if not, an error message
  is returned. The output is a quadratic form in the category [10XIsQuadraticForm[0m.
  (See [14X3.2[0m for more on quadratic forms).
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> r := PolynomialRing( GF(8), 3);[0X
    [4XGF(2^3)[x_1,x_2,x_3][0X
    [4Xgap> poly := r.1^2 + r.2^2 + r.3^2;[0X
    [4Xx_1^2+x_2^2+x_3^2[0X
    [4Xgap> form := QuadraticFormByPolynomial(poly, r);[0X
    [4X< quadratic form >[0X
    [4Xgap> RadicalOfForm(form);[0X
    [4X<vector space over GF(2^3), with 63 generators>[0X
    [4Xgap> r := PolynomialRing(GF(9),4);[0X
    [4XGF(3^2)[x_1,x_2,x_3,x_4][0X
    [4Xgap> poly := Z(3)^2*r.1^2+r.2^2+r.3*r.4;[0X
    [4Xx_1^2+x_2^2+x_3*x_4[0X
    [4Xgap> qform := QuadraticFormByPolynomial(poly,r);[0X
    [4X< quadratic form >[0X
    [4Xgap> Display(qform);[0X
    [4XQuadratic form[0X
    [4XGram Matrix:[0X
    [4X 1 . . .[0X
    [4X . 1 . .[0X
    [4X . . . 1[0X
    [4X . . . .[0X
    [4XPolynomial: x_1^2+x_2^2+x_3*x_4[0X
    [4X [0X
  [4X------------------------------------------------------------------[0X
  
  [1X4.3-3 HermitianFormByPolynomial[0m
  
  [2X> HermitianFormByPolynomial( [0X[3Xpoly, r, n[0X[2X ) _________________________[0Xoperation
  [2X> HermitianFormByPolynomial( [0X[3Xpoly, r[0X[2X ) ____________________________[0Xoperation
  [6XReturns:[0X  an hermitian form
  
  The argument [3Xpoly[0m must be a polynomial in the polynomial ring [3Xr[0m defined over
  a  finite  field  of  square  order  q^2 The (optional) last argument is the
  dimension  for  the  underlying vector space of the resulting form, which by
  default  is  the  number  of indeterminates specified by [3Xpoly[0m. It is checked
  whether  the  polynomial  is  a homogeneous polynomial of degree q+1, and if
  not,  an  error  message  is returned. The output is a hermitian form in the
  category [10XIsHermitianForm[0m. (See [14X3.1[0m for more on hermitian forms).
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> r := PolynomialRing( GF(9), 4);[0X
    [4XGF(3^2)[x_1,x_2,x_3,x_4][0X
    [4Xgap> vars := IndeterminatesOfPolynomialRing( r );[0X
    [4X[ x_1, x_2, x_3, x_4 ][0X
    [4Xgap> poly := vars[1]*vars[2]^3+vars[1]^3*vars[2]+[0X
    [4X>              vars[3]*vars[4]^3+vars[3]^3*vars[4];[0X
    [4Xx_1^3*x_2+x_1*x_2^3+x_3^3*x_4+x_3*x_4^3[0X
    [4Xgap> form := HermitianFormByPolynomial(poly,r);[0X
    [4X< hermitian form >[0X
    [4Xgap> Display(form);[0X
    [4XHermitian form[0X
    [4XGram Matrix:[0X
    [4X . 1 . .[0X
    [4X 1 . . .[0X
    [4X . . . 1[0X
    [4X . . 1 .[0X
    [4XPolynomial: x_1^3*x_2+x_1*x_2^3+x_3^3*x_4+x_3*x_4^3[0X
    [4X [0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X4.4 Switching between bilinear and quadratic forms[0X
  
  When  the  characteristic  of  the  field  is  odd,  a homogeneous quadratic
  polynomial  determines  a  bilinear  form,  and  a  quadratic  form. In some
  situations,  when  a quadratic form Q is given, it is useful to consider the
  bilinear  form  f  such  that  f(v,v)=Q(v), i.e., the bilinear form which is
  determined  by exactly the same polynomial determining the quadratic form Q.
  [5XForms[0m  provides  functionality  to  construct a bilinear form f from a given
  quadratic  form  Q  such  that  f(v,v)=Q(v).  Conversely,  we  can extract a
  quadratic form from a given bilinear form.
  
  [1X4.4-1 QuadraticFormByBilinearForm[0m
  
  [2X> QuadraticFormByBilinearForm( [0X[3Xform[0X[2X ) _____________________________[0Xoperation
  [6XReturns:[0X  a quadratic form
  
  The  argument  form  is  an orthogonal bilinear form (and thus it belongs to
  [10XIsBilinearForm[0m),  otherwise  a  ``No  method  found'' error is returned. The
  output  is  the  quadratic  form Q (an object in [10XIsQuadraticForm[0m), such that
  Q(v)  = form(v,v) for all vectors v in a vector space equipped with form. An
  error is returned when the characteristic of the field is even, or when form
  is not orthogonal.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> mat := [ [ Z(3^2)^7, Z(3)^0, Z(3^2)^2, 0*Z(3), Z(3^2)^5 ], [0X
    [4X>    [ Z(3)^0, Z(3^2)^7, Z(3^2)^6, Z(3^2)^5, Z(3^2)^2 ], [0X
    [4X>    [ Z(3^2)^2, Z(3^2)^6, Z(3^2)^7, Z(3^2)^2, Z(3^2)^2 ], [0X
    [4X>    [ 0*Z(3), Z(3^2)^5, Z(3^2)^2, Z(3^2)^6, Z(3^2)^7 ], [0X
    [4X>    [ Z(3^2)^5, Z(3^2)^2, Z(3^2)^2, Z(3^2)^7, Z(3) ] ];[0X
    [4X[ [ Z(3^2)^7, Z(3)^0, Z(3^2)^2, 0*Z(3), Z(3^2)^5 ], [0X
    [4X  [ Z(3)^0, Z(3^2)^7, Z(3^2)^6, Z(3^2)^5, Z(3^2)^2 ], [0X
    [4X  [ Z(3^2)^2, Z(3^2)^6, Z(3^2)^7, Z(3^2)^2, Z(3^2)^2 ], [0X
    [4X  [ 0*Z(3), Z(3^2)^5, Z(3^2)^2, Z(3^2)^6, Z(3^2)^7 ], [0X
    [4X  [ Z(3^2)^5, Z(3^2)^2, Z(3^2)^2, Z(3^2)^7, Z(3) ] ][0X
    [4Xgap> form := BilinearFormByMatrix(mat,GF(9));[0X
    [4X< bilinear form >[0X
    [4Xgap> Q := QuadraticFormByBilinearForm(form);[0X
    [4X< quadratic form >[0X
    [4Xgap> Display(form);[0X
    [4XBilinear form[0X
    [4XGram Matrix:[0X
    [4Xz = Z(9)[0X
    [4X z^7   1 z^2   . z^5[0X
    [4X   1 z^7 z^6 z^5 z^2[0X
    [4X z^2 z^6 z^7 z^2 z^2[0X
    [4X   . z^5 z^2 z^6 z^7[0X
    [4X z^5 z^2 z^2 z^7   2[0X
    [4Xgap> Display(Q);[0X
    [4XQuadratic form[0X
    [4XGram Matrix:[0X
    [4Xz = Z(9)[0X
    [4X z^7   2 z^6   . z^1[0X
    [4X   . z^7 z^2 z^1 z^6[0X
    [4X   .   . z^7 z^6 z^6[0X
    [4X   .   .   . z^6 z^3[0X
    [4X   .   .   .   .   2[0X
    [4Xgap> Set(List(GF(9)^5),x->[x,x]^form=x^Q);[0X
    [4X[ true ][0X
    [4Xgap> PolynomialOfForm(form);[0X
    [4XZ(3^2)^7*x_1^2-x_1*x_2+Z(3^2)^6*x_1*x_3+Z(3^2)*x_1*x_5+Z(3^2)^7*x_2^2+Z(3^2)^2[0X
    [4X*x_2*x_3+Z(3^2)*x_2*x_4+Z(3^2)^6*x_2*x_5+Z(3^2)^7*x_3^2+Z(3^2)^6*x_3*x_4+Z(3^2[0X
    [4X)^6*x_3*x_5+Z(3^2)^6*x_4^2+Z(3^2)^3*x_4*x_5-x_5^2[0X
    [4Xgap> PolynomialOfForm(Q);[0X
    [4XZ(3^2)^7*x_1^2-x_1*x_2+Z(3^2)^6*x_1*x_3+Z(3^2)*x_1*x_5+Z(3^2)^7*x_2^2+Z(3^2)^2[0X
    [4X*x_2*x_3+Z(3^2)*x_2*x_4+Z(3^2)^6*x_2*x_5+Z(3^2)^7*x_3^2+Z(3^2)^6*x_3*x_4+Z(3^2[0X
    [4X)^6*x_3*x_5+Z(3^2)^6*x_4^2+Z(3^2)^3*x_4*x_5-x_5^2[0X
    [4X [0X
  [4X------------------------------------------------------------------[0X
  
  Note  that  the given bilinear form [3Xform[0m is [12Xnot[0m the associated bilinear form
  of  the constructed quadratic form Q, according to the definition in Section
  [14X3.2[0m.   We   can   construct   the   associated   bilinear   forms  by  using
  [2XAssociatedBilinearForm[0m ([14X4.4-3[0m). (See [14X3.2[0m for more on quadratic forms).
  
  [1X4.4-2 BilinearFormByQuadraticForm[0m
  
  [2X> BilinearFormByQuadraticForm( [0X[3XQ[0X[2X ) ________________________________[0Xoperation
  [6XReturns:[0X  a bilinear form
  
  The   argument  Q  must  be  a  quadratic  form  (and  thus  it  belongs  to
  [10XIsQuadraticForm[0m). The output is the orthogonal bilinear form f (an object in
  [10XIsBilinearForm[0m), such that f(v,v) = Q(v) for all vectors v in a vector space
  equipped  with  Q. An error is returned when the characteristic of the field
  is even.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> r := PolynomialRing(GF(9),4);[0X
    [4XGF(3^2)[x_1,x_2,x_3,x_4][0X
    [4Xgap> poly := -r.1*r.2+Z(3^2)*r.3^2+r.4^2;[0X
    [4X-x_1*x_2+Z(3^2)*x_3^2+x_4^2[0X
    [4Xgap> qform := QuadraticFormByPolynomial(poly,r);[0X
    [4X< quadratic form >[0X
    [4Xgap> Display( qform );[0X
    [4XQuadratic form[0X
    [4XGram Matrix:[0X
    [4Xz = Z(9)[0X
    [4X   .   2   .   .[0X
    [4X   .   .   .   .[0X
    [4X   .   . z^1   .[0X
    [4X   .   .   .   1[0X
    [4XPolynomial: -x_1*x_2+Z(3^2)*x_3^2+x_4^2[0X
    [4Xgap> form := BilinearFormByQuadraticForm( qform );[0X
    [4X< bilinear form >[0X
    [4Xgap> Display(form);[0X
    [4XBilinear form[0X
    [4XGram Matrix:[0X
    [4Xz = Z(9)[0X
    [4X   .   1   .   .[0X
    [4X   1   .   .   .[0X
    [4X   .   . z^1   .[0X
    [4X   .   .   .   1[0X
    [4Xgap> Set(GF(9)^4, x -> [x,x]^form = x^qform);[0X
    [4X[ true ][0X
    [4X [0X
  [4X------------------------------------------------------------------[0X
  
  Note  that  the  constructed  bilinear form f is [12Xnot[0m the associated bilinear
  form  of  the given quadratic form Q, according to the definition in Section
  [14X3.2[0m.   We   can   construct   the   associated   bilinear   forms  by  using
  [2XAssociatedBilinearForm[0m ([14X4.4-3[0m). (See [14X3.2[0m for more on quadratic forms).
  
  [1X4.4-3 AssociatedBilinearForm[0m
  
  [2X> AssociatedBilinearForm( [0X[3XQ[0X[2X ) _____________________________________[0Xoperation
  [6XReturns:[0X  a bilinear form
  
  The   argument  Q  must  be  a  quadratic  form  (and  thus  it  belongs  to
  [10XIsQuadraticForm[0m). The output is the associated bilinear form f (an object in
  [10XIsBilinearForm[0m),  as  defined  in Section [14X3.2[0m, i.e. the bilinear form f such
  that  f(v,w)  =  Q(v+w)-Q(v)-Q(w)  for  all  vectors  v,w  in a vector space
  equipped with Q. (See [14X3.2[0m for more on quadratic forms).
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> r:= PolynomialRing(GF(121),6);[0X
    [4XGF(11^2)[x_1,x_2,x_3,x_4,x_5,x_6][0X
    [4Xgap> poly := r.1*r.5-r.2*r.6+r.3*r.4;[0X
    [4Xx_1*x_5-x_2*x_6+x_3*x_4[0X
    [4Xgap> form := QuadraticFormByPolynomial(poly,r);[0X
    [4X< quadratic form >[0X
    [4Xgap> aform := AssociatedBilinearForm(form);[0X
    [4X< bilinear form >[0X
    [4Xgap> Display(aform);[0X
    [4XBilinear form[0X
    [4XGram Matrix:[0X
    [4X  .  .  .  .  1  .[0X
    [4X  .  .  .  .  . 10[0X
    [4X  .  .  .  1  .  .[0X
    [4X  .  .  1  .  .  .[0X
    [4X  1  .  .  .  .  .[0X
    [4X  . 10  .  .  .  .[0X
    [4X [0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X4.5 Evaluating forms[0X
  
  [1X4.5-1 EvaluateForm[0m
  
  [2X> EvaluateForm( [0X[3Xf, u, v[0X[2X ) _________________________________________[0Xoperation
  [2X> EvaluateForm( [0X[3Xf, u[0X[2X ) ____________________________________________[0Xoperation
  [6XReturns:[0X  a finite field element
  
  The  argument  [3Xf[0m  is  either a sesquilinear or quadratic form defined over a
  finite  field GF(q). The other argument is a pair of vectors or matrices, or
  a  single  vector or matrix, which represent the bases of given subspaces of
  GF(q)^d.  This  operation  evaluates the form on the given vector or pair of
  vectors and returns an element in GF(q). There is also an overloading of the
  operation  [3X\^[0m  where  (u,v)^f  represents  f(u,v)  in  the  case  that  [3Xf[0m is
  sesquilinear,  and  u^f  stands  for  f(u)  in  the  quadratic  case. So for
  convenience,  the  user  may  use this compressed version of this operation,
  which we show in the following example:
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> mat := [[Z(8),0,0,0],[0,0,Z(8)^4,0],[0,0,0,1],[0,0,0,0]]*Z(8)^0;;[0X
    [4Xgap> form := QuadraticFormByMatrix(mat,GF(8));[0X
    [4X< quadratic form >[0X
    [4Xgap> u := [ Z(2^3)^4, Z(2^3)^4, Z(2)^0, Z(2^3)^3 ];[0X
    [4X[ Z(2^3)^4, Z(2^3)^4, Z(2)^0, Z(2^3)^3 ][0X
    [4Xgap> EvaluateForm( form, u );[0X
    [4XZ(2^3)^6[0X
    [4Xgap> u^form;[0X
    [4XZ(2^3)^6[0X
    [4Xgap> gram := [[0,0,0,0,0,2],[0,0,0,0,2,0],[0,0,0,1,0,0],[0X
    [4X>               [0,0,1,0,0,0],[0,2,0,0,0,0],[2,0,0,0,0,0]]*Z(3)^0;;[0X
    [4Xgap> form := BilinearFormByMatrix(gram,GF(3));[0X
    [4X< bilinear form >[0X
    [4Xgap> u := [ [ Z(3)^0, 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0 ], [0X
    [4X>   [ 0*Z(3), 0*Z(3), Z(3)^0, Z(3)^0, Z(3), 0*Z(3) ] ];;[0X
    [4Xgap> v := [ [ Z(3)^0, 0*Z(3), Z(3)^0, Z(3), 0*Z(3), Z(3) ], [0X
    [4X>   [ 0*Z(3), Z(3)^0, 0*Z(3), Z(3), Z(3), Z(3) ] ];;[0X
    [4Xgap> EvaluateForm( form, u, v);[0X
    [4X[ [ Z(3)^0, Z(3)^0 ], [ 0*Z(3), 0*Z(3) ] ][0X
    [4Xgap> [u,v]^form;[0X
    [4X[ [ Z(3)^0, Z(3)^0 ], [ 0*Z(3), 0*Z(3) ] ][0X
    [4X [0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X4.6   Orthogonality,  totally  isotropic  subspaces,  and  totally  singular[0X
  [1Xsubspaces[0X
  
  [1X4.6-1 OrthogonalSubspaceMat[0m
  
  [2X> OrthogonalSubspaceMat( [0X[3Xform, v[0X[2X ) ________________________________[0Xoperation
  [6XReturns:[0X  a  base  of  the  subspace  orthogonal  to  the  given vector with
            relation to the given form
  
  The argument [3Xform[0m is a sesquilinear or quadratic form. The operation returns
  a base of the subspace orthogonal to the given vector [3Xv[0m with relation to the
  sesquilinear  [3Xform[0m  or  with relation to the associated bilinear form of the
  quadratic form [3Xform[0m
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> mat := [[0,0,0,-2],[0,0,-3,0],[0,3,0,0],[2,0,0,0]]*Z(7)^0;[0X
    [4X[ [ 0*Z(7), 0*Z(7), 0*Z(7), Z(7)^5 ], [ 0*Z(7), 0*Z(7), Z(7)^4, 0*Z(7) ], [0X
    [4X  [ 0*Z(7), Z(7), 0*Z(7), 0*Z(7) ], [ Z(7)^2, 0*Z(7), 0*Z(7), 0*Z(7) ] ][0X
    [4Xgap> form := BilinearFormByMatrix(mat);[0X
    [4X< bilinear form >[0X
    [4Xgap> v := Random(GF(7)^4);[0X
    [4X[ Z(7)^3, Z(7)^2, Z(7)^4, Z(7) ][0X
    [4Xgap> vperp := OrthogonalSubspaceMat(form,v);[0X
    [4X[ [ Z(7)^5, Z(7)^0, 0*Z(7), 0*Z(7) ], [ Z(7)^0, 0*Z(7), Z(7)^0, 0*Z(7) ], [0X
    [4X  [ Z(7)^2, 0*Z(7), 0*Z(7), Z(7)^0 ] ][0X
    [4Xgap> sub := [[1,1,0,0],[0,0,1,2]]*Z(7)^0;[0X
    [4X[ [ Z(7)^0, Z(7)^0, 0*Z(7), 0*Z(7) ], [ 0*Z(7), 0*Z(7), Z(7)^0, Z(7)^2 ] ][0X
    [4Xgap> subperp := OrthogonalSubspaceMat(form,sub);[0X
    [4X[ [ Z(7)^0, Z(7)^0, 0*Z(7), 0*Z(7) ], [ 0*Z(7), 0*Z(7), Z(7)^4, Z(7)^0 ] ][0X
    [4Xgap> mat := [[1,0,0],[0,0,1],[0,0,0]]*Z(2)^0;[0X
    [4X[ [ Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), Z(2)^0 ], [0X
    [4X  [ 0*Z(2), 0*Z(2), 0*Z(2) ] ][0X
    [4Xgap> form := QuadraticFormByMatrix(mat);[0X
    [4X< quadratic form >[0X
    [4Xgap> v := Random(GF(2)^3);[0X
    [4X<a GF2 vector of length 3>[0X
    [4Xgap> vperp := OrthogonalSubspaceMat(form,v);[0X
    [4X[ <an immutable GF2 vector of length 3>, <an immutable GF2 vector of length [0X
    [4X    3> ][0X
    [4Xgap> sub := [[1,0,1],[1,0,0]]*Z(2)^0;[0X
    [4X[ [ Z(2)^0, 0*Z(2), Z(2)^0 ], [ Z(2)^0, 0*Z(2), 0*Z(2) ] ][0X
    [4Xgap> subperp := OrthogonalSubspaceMat(form,sub);[0X
    [4X[ <an immutable GF2 vector of length 3>, <an immutable GF2 vector of length [0X
    [4X    3> ][0X
    [4X [0X
  [4X------------------------------------------------------------------[0X
  
  [1X4.6-2 IsIsotropicVector[0m
  
  [2X> IsIsotropicVector( [0X[3Xform, v[0X[2X ) ____________________________________[0Xoperation
  [6XReturns:[0X  true or false
  
  The operation return [3Xtrue[0m if and only if [3Xv[0m is isotropic with relation to the
  sesquilinear or quadratic form [3Xform[0m.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> mat := [[1,0,0,0],[0,-1,0,0],[0,0,0,1],[0,0,1,0]]*Z(41)^0;[0X
    [4X[ [ Z(41)^0, 0*Z(41), 0*Z(41), 0*Z(41) ], [0X
    [4X  [ 0*Z(41), Z(41)^20, 0*Z(41), 0*Z(41) ], [0X
    [4X  [ 0*Z(41), 0*Z(41), 0*Z(41), Z(41)^0 ], [0X
    [4X  [ 0*Z(41), 0*Z(41), Z(41)^0, 0*Z(41) ] ][0X
    [4Xgap> form := BilinearFormByMatrix(mat);[0X
    [4X< bilinear form >[0X
    [4Xgap> v := [1,1,0,0]*Z(41)^0;[0X
    [4X[ Z(41)^0, Z(41)^0, 0*Z(41), 0*Z(41) ][0X
    [4Xgap> IsIsotropicVector(form,v);[0X
    [4Xtrue[0X
    [4Xgap> mat := [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,0,0],[0,0,1,0,0],[0,0,0,0,0]]*Z(8)^0;[0X
    [4X[ [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [0X
    [4X  [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], [0X
    [4X  [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [0X
    [4X  [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], [0X
    [4X  [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ][0X
    [4Xgap> form := QuadraticFormByMatrix(mat);[0X
    [4X< quadratic form >[0X
    [4Xgap> v1 := [1,0,0,0,0]*Z(8)^0;[0X
    [4X[ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ][0X
    [4Xgap> v2 := [0,1,0,0,0]*Z(8)^0;[0X
    [4X[ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ][0X
    [4Xgap> IsIsotropicVector(form,v1);[0X
    [4Xtrue[0X
    [4Xgap> IsIsotropicVector(form,v2);[0X
    [4Xtrue[0X
    [4X [0X
  [4X------------------------------------------------------------------[0X
  
  [1X4.6-3 IsSingularVector[0m
  
  [2X> IsSingularVector( [0X[3Xform, v[0X[2X ) _____________________________________[0Xoperation
  [6XReturns:[0X  true or false
  
  The  operation return [3Xtrue[0m if and only if [3Xv[0m is singular with relation to the
  quadratic  form [3Xform[0m. Note that only when the characteristic of the field is
  odd,  the  singular  vectors  with  relation  to  a  quadratic  form are the
  isotropic vectors with relation to its associated form.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> mat := [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,0,0],[0,0,1,0,0],[0,0,0,0,0]]*Z(8)^0;[0X
    [4X[ [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [0X
    [4X  [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], [0X
    [4X  [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [0X
    [4X  [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], [0X
    [4X  [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ][0X
    [4Xgap> form := QuadraticFormByMatrix(mat);[0X
    [4X< quadratic form >[0X
    [4Xgap> v1 := [1,0,0,0,0]*Z(8)^0;[0X
    [4X[ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ][0X
    [4Xgap> v2 := [0,1,0,0,0]*Z(8)^0;[0X
    [4X[ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ][0X
    [4Xgap> IsSingularVector(form,v1);[0X
    [4Xfalse[0X
    [4Xgap> IsSingularVector(form,v2);[0X
    [4Xtrue[0X
    [4Xgap> IsIsotropicVector(form,v1);[0X
    [4Xtrue[0X
    [4Xgap> IsIsotropicVector(form,v2);[0X
    [4Xtrue[0X
    [4X [0X
  [4X------------------------------------------------------------------[0X
  
  [1X4.6-4 IsTotallyIsotropicSubspace[0m
  
  [2X> IsTotallyIsotropicSubspace( [0X[3Xform, sub[0X[2X ) _________________________[0Xoperation
  [6XReturns:[0X  true or false
  
  The operation return [3Xtrue[0m if and only if the subspace spanned by the vectors
  in  the  list  [3Xsub[0m is totally isotropic with relation to the sesquilinear or
  quadratic  form [3Xform[0m. Note that when [3Xform[0m is a quadratic form, it is checked
  whether  [3Xsub[0m generates a subspace that is totally isotropic with relation to
  the associated bilinear form of [3Xform[0m.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> mat := [[1,0,0,0],[0,-1,0,0],[0,0,0,1],[0,0,1,0]]*Z(7)^0;[0X
    [4X[ [ Z(7)^0, 0*Z(7), 0*Z(7), 0*Z(7) ], [ 0*Z(7), Z(7)^3, 0*Z(7), 0*Z(7) ], [0X
    [4X  [ 0*Z(7), 0*Z(7), 0*Z(7), Z(7)^0 ], [ 0*Z(7), 0*Z(7), Z(7)^0, 0*Z(7) ] ][0X
    [4Xgap> form := BilinearFormByMatrix(mat);[0X
    [4X< bilinear form >[0X
    [4Xgap> sub:= [[Z(7)^0,0*Z(7),Z(7)^0,Z(7)],[0*Z(7),Z(7)^0,Z(7)^0,Z(7)^4]];[0X
    [4X[ [ Z(7)^0, 0*Z(7), Z(7)^0, Z(7) ], [ 0*Z(7), Z(7)^0, Z(7)^0, Z(7)^4 ] ][0X
    [4Xgap> IsTotallyIsotropicSubspace(form,sub);[0X
    [4Xtrue[0X
    [4Xgap> mat := IdentityMat(6,GF(2));[0X
    [4X[ <a GF2 vector of length 6>, <a GF2 vector of length 6>, [0X
    [4X  <a GF2 vector of length 6>, <a GF2 vector of length 6>, [0X
    [4X  <a GF2 vector of length 6>, <a GF2 vector of length 6> ][0X
    [4Xgap> form := HermitianFormByMatrix(mat,GF(4));[0X
    [4X< hermitian form >[0X
    [4Xgap> sub := [[Z(2)^0,0*Z(2),0*Z(2),Z(2)^0,Z(2)^0,Z(2)^0], [0X
    [4X>   [0*Z(2),Z(2)^0,0*Z(2),Z(2^2)^2,Z(2^2),Z(2)^0], [0X
    [4X>   [0*Z(2),0*Z(2),Z(2)^0,Z(2)^0,Z(2^2),Z(2^2)^2]];[0X
    [4X[ [ Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0 ], [0X
    [4X  [ 0*Z(2), Z(2)^0, 0*Z(2), Z(2^2)^2, Z(2^2), Z(2)^0 ], [0X
    [4X  [ 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, Z(2^2), Z(2^2)^2 ] ][0X
    [4Xgap> IsTotallyIsotropicSubspace(form,sub);[0X
    [4Xtrue[0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  [1X4.6-5 IsTotallySingularSubspace[0m
  
  [2X> IsTotallySingularSubspace( [0X[3Xform, sub[0X[2X ) __________________________[0Xoperation
  [6XReturns:[0X  true or false
  
  The operation return [3Xtrue[0m if and only if the subspace spanned by the vectors
  in  the  list  [3Xsub[0m is totally singular with relation to quadratic form [3Xform[0m.
  Note  that  only  when  the  characteristic of the field is odd, the totally
  singular  subspaces  of  given dimension n with relation to a quadratic form
  are  exactly the totally isotropic subspaces of dimension n with relation to
  its associated form.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> mat := [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,0,0],[0,0,1,0,0],[0,0,0,0,0]]*Z(8)^0;[0X
    [4X[ [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [0X
    [4X  [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], [0X
    [4X  [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [0X
    [4X  [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], [0X
    [4X  [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ][0X
    [4Xgap> form := QuadraticFormByMatrix(mat);[0X
    [4X< quadratic form >[0X
    [4Xgap> sub := [[Z(2)^0,0*Z(2),Z(2^3)^6,Z(2^3),Z(2^3)^3],[0X
    [4X>        [0*Z(2),Z(2)^0,Z(2^3)^6,Z(2^3)^2,Z(2^3)]];[0X
    [4X[ [ Z(2)^0, 0*Z(2), Z(2^3)^6, Z(2^3), Z(2^3)^3 ], [0X
    [4X  [ 0*Z(2), Z(2)^0, Z(2^3)^6, Z(2^3)^2, Z(2^3) ] ][0X
    [4Xgap> IsTotallySingularSubspace(form,sub);[0X
    [4Xtrue[0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X4.7 Attributes and properties of forms[0X
  
  [1X4.7-1 IsReflexiveForm[0m
  
  [2X> IsReflexiveForm( [0X[3Xf[0X[2X ) _____________________________________________[0Xproperty
  [6XReturns:[0X  true or false.
  
  A  sesquilinear  form  f  on  a  vector  space V is [13Xreflexive[0m if f(v,w)=0 =>
  f(w,v)=0  for  all v,w in V. The argument f must be a sesquilinear form (and
  thus  it  belongs to [10XIsSesquilinearForm[0m). A sesquilinear form f is [13Xreflexive[0m
  if  whenever  we have f(u,v)=0, for two vectors u,v in the associated vector
  space,  then  we  also  have f(v,u)=0. This attribute simply returns [3Xtrue[0m or
  [3Xfalse[0m  according  to  whether  [3Xf[0m  is  reflexive  or  not, and is stored as a
  property  of  [3Xf[0m.  It  is  not possible in this version of [5XForms[0m to construct
  non-reflexive forms. (See [14X3.1[0m for more on reflexive sesquilinear forms).
  
  [1X4.7-2 IsAlternatingForm[0m
  
  [2X> IsAlternatingForm( [0X[3Xf[0X[2X ) ___________________________________________[0Xproperty
  [6XReturns:[0X  true or false.
  
  A sesquilinear form f on a vector space V is [13Xalternating[0m if f(v,v)=0 for all
  v  in  V. The argument f must be a sesquilinear form (and thus it belongs to
  [10XIsSesquilinearForm[0m). A bilinear form f is [13Xalternating[0m if f(v,v)=0 for all v.
  This  method  simply  returns  [3Xtrue[0m  or  [3Xfalse[0m  according  to  whether  [3Xf[0m is
  alternating  or  not, and is stored as a property of [3Xf[0m. (See [14X3.1[0m for more on
  alternating sesquilinear forms).
  
  [1X4.7-3 IsSymmetricForm[0m
  
  [2X> IsSymmetricForm( [0X[3Xf[0X[2X ) _____________________________________________[0Xproperty
  [6XReturns:[0X  true or false.
  
  A  sesquilinear form f on a vector space V is [13Xsymmetric[0m if f(v,w)=f(w,v) for
  all  v,w  in  V.  The  argument  f  must be a sesquilinear form (and thus it
  belongs   to   [10XIsSesquilinearForm[0m).  A  bilinear  form  f  is  [13Xsymmetric[0m  if
  f(u,v)=f(v,u)  for  all  pairs  of  vectors  u  and v. This attribute simply
  returns  [3Xtrue[0m  or  [3Xfalse[0m  according to whether [3Xf[0m is symmetric or not, and is
  stored  as  a  property  of  [3Xf[0m.  (See [14X3.1[0m for more on symmetric sesquilinear
  forms).
  
  [1X4.7-4 IsOrthogonalForm[0m
  
  [2X> IsOrthogonalForm( [0X[3Xf[0X[2X ) ____________________________________________[0Xproperty
  [6XReturns:[0X  true or false.
  
  The  argument  f  must  be  a  sesquilinear  form  (and  thus  it belongs to
  [10XIsSesquilinearForm[0m).   A  bilinear  form  f  is  called  [13Xorthogonal[0m  if  the
  characteristic  of  the  underlying field is odd, and f is a symmetric form.
  (See  [14X3.1[0m for more on bilinear forms). This operation simply returns [3Xtrue[0m or
  [3Xfalse[0m  according  to whether [3Xf[0m is an orthogonal bilinear form or not, and is
  stored as a property of [3Xf[0m.
  
  [1X4.7-5 IsPseudoForm[0m
  
  [2X> IsPseudoForm( [0X[3Xf[0X[2X ) ________________________________________________[0Xproperty
  [6XReturns:[0X  true or false.
  
  When  the characteristic of the field is odd, we call a form f [13Xorthogonal[0m if
  and  only  f is symmetric, and when the characteristic of the field is even,
  we  call  a form f [13Xpseudo[0m if and only if f is symmetric but not alternating.
  The  argument  f  must  be  a  sesquilinear  form  (and  thus  it belongs to
  [10XIsSesquilinearForm[0m).  (See [14X3.1[0m for more on pseudo forms). This method simply
  returns [3Xtrue[0m or [3Xfalse[0m according to whether [3Xf[0m is a pseudo form or not, and is
  stored as a property of [3Xf[0m.
  
  [1X4.7-6 IsSymplecticForm[0m
  
  [2X> IsSymplecticForm( [0X[3Xf[0X[2X ) ____________________________________________[0Xproperty
  [6XReturns:[0X  true or false.
  
  We  call  a  bilinear form f [13Xsymplectic[0m if and only if f is alternating. The
  argument   f   must   be  a  sesquilinear  form  (and  thus  it  belongs  to
  [10XIsSesquilinearForm[0m).  (See  [14X3.1[0m  for  more on symplectic forms). This method
  simply  returns  [3Xtrue[0m  or [3Xfalse[0m according to whether [3Xf[0m is symplectic or not,
  and is stored as a property of [3Xf[0m.
  
  [1X4.7-7 IsDegenerateForm[0m
  
  [2X> IsDegenerateForm( [0X[3Xf[0X[2X ) ____________________________________________[0Xproperty
  [6XReturns:[0X  true or false.
  
  The  argument  f  must  be  a  form  (and  thus  it  belongs  to  [10XIsForm[0m). A
  sesquilinear form f is [13Xdegenerate[0m if its radical is non-trivial. A quadratic
  form  is  degenerate  if  and only if the radical of the associated bilinear
  form  is  non-trivial.  Note  that  degeneracy  for  quadratic  forms is too
  restrictive  if the characteristic is even. See also [2XIsSingularForm[0m ([14X4.7-8[0m).
  This  attribute  simply  returns  [3Xtrue[0m  or  [3Xfalse[0m  according to whether [3Xf[0m is
  degenerate or not, and is stored as a property of [3Xf[0m.
  
  [1X4.7-8 IsSingularForm[0m
  
  [2X> IsSingularForm( [0X[3Xf[0X[2X ) ______________________________________________[0Xproperty
  [6XReturns:[0X  true or false.
  
  The   argument  f  must  be  a  quadratic  form  (and  thus  it  belongs  to
  [10XIsQuadraticForm[0m).  A  quadratic  form  f  is  [13Xsingular[0m  if  its  radical  is
  non-trivial.  When  the characteristic of the field is odd, a quadratic form
  is  singular  if and only if it is degenerate. This is not the case when the
  characteristic  of  the  field  is  even. This method simply returns [3Xtrue[0m or
  [3Xfalse[0m according to whether [3Xf[0m is singular or not, and is stored as a property
  of [3Xf[0m.
  
  [1X4.7-9 BaseField[0m
  
  [2X> BaseField( [0X[3Xf[0X[2X ) __________________________________________________[0Xattribute
  [6XReturns:[0X  the underlying field of [3Xf[0m.
  
  The  argument  f  must be a form (and thus it belongs to [10XIsForm[0m). The method
  returns  the  field which is stored as the [13Xdefining field[0m of f. We sometimes
  stipulate  in  [5XForms[0m  that  a  form  have a defining field, for mathematical
  reasons.  Clearly, to define a hermitian form one needs to specify the field
  of scalars for the vector space that you wish your hermitian form to act on.
  The default, if the user has not specified a field on creation of a form, is
  the  smallest  field  containing the entries or coefficients of the input (a
  matrix  or polynomial). Having a particular defining field for a form can be
  very  useful, for example, when one wants to find a change of basis from one
  form  to  another (isometric) form. In this case, one needs to know in which
  GL(d,q) the base-transition matrix should be taken.
  
  [1X4.7-10 GramMatrix[0m
  
  [2X> GramMatrix( [0X[3Xf[0X[2X ) _________________________________________________[0Xattribute
  [6XReturns:[0X  the Gram matrix of [3Xf[0m.
  
  The  argument  f must be a form (and thus it belongs to [10XIsForm[0m). This method
  returns the Gram matrix of f (see [14X3.1[0m and [14X3.2[0m).
  
  [1X4.7-11 RadicalOfForm[0m
  
  [2X> RadicalOfForm( [0X[3Xf[0X[2X ) ______________________________________________[0Xattribute
  [6XReturns:[0X  The radical of the form [3Xf[0m
  
  The argument f must be a form (and thus it belongs to [10XIsForm[0m) on some vector
  space V. The radical of a form f is the subspace consisting of vectors which
  are orthogonal to every vector, i.e.,
  
  
       Rad(f) = \{v \in V | f(v,w) = 0,\, \forall w \in V\}.
  
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> r := PolynomialRing( GF(8), 3 );[0X
    [4XGF(2^3)[x_1,x_2,x_3][0X
    [4Xgap> poly := r.1^2 + r.2 * r.3;[0X
    [4Xx_1^2+x_2*x_3[0X
    [4Xgap> form := QuadraticFormByPolynomial( poly, r );[0X
    [4X< quadratic form >[0X
    [4Xgap> r := RadicalOfForm( form );[0X
    [4X<vector space over GF(2^3), with 0 generators>[0X
    [4Xgap> Dimension(r);[0X
    [4X0[0X
    [4X [0X
  [4X------------------------------------------------------------------[0X
  
  [1X4.7-12 PolynomialOfForm[0m
  
  [2X> PolynomialOfForm( [0X[3Xf[0X[2X ) ___________________________________________[0Xattribute
  [6XReturns:[0X  the polynomial associated with [3Xf[0m.
  
  The  argument  [3Xf[0m  must be a form (and thus it belongs to [10XIsForm[0m). All forms,
  except  for  bilinear  forms  in  even  characteristic,  have  an associated
  polynomial  defining  a  quadratic or hermitian form (see [14X3.1[0m and [14X3.2[0m). This
  method  returns  the polynomial associated with [3Xf[0m, and if not already bound,
  stores it as a property of [3Xf[0m.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> mat := [ [ Z(8) , 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [0X
    [4X>  [ 0*Z(2), Z(2)^0, Z(2^3)^5, 0*Z(2), 0*Z(2) ], [0X
    [4X>  [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [0X
    [4X>  [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], [0X
    [4X>  [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ];;[0X
    [4Xgap> form := QuadraticFormByMatrix(mat,GF(8));[0X
    [4X< quadratic form >[0X
    [4Xgap> PolynomialOfForm(form);[0X
    [4XZ(2^3)*x_1^2+x_2^2+Z(2^3)^5*x_2*x_3+x_4*x_5[0X
    [4X [0X
  [4X------------------------------------------------------------------[0X
  
  [1X4.7-13 DiscriminantOfForm[0m
  
  [2X> DiscriminantOfForm( [0X[3Xf[0X[2X ) _________________________________________[0Xattribute
  [6XReturns:[0X  a string
  
  The  argument  f  must  be  a  form (and thus it belongs to [10XIsForm[0m). Given a
  quadratic  or  bilinear  form  [3Xf[0m of even dimension, this operation returns a
  string: ``square'' or ``nonsquare''. More specifically, let f be a from over
  GF(q),  and  let  M  be  the  Gram matrix of f. Define the [13Xdiscriminant[0m of Q
  (n.b.,  [13Xquasideterminant[0m  in [CCNPW85]) as `square' if det(M) is a square of
  GF(q),  and  `non-square'  otherwise.  The  discriminant  is an invariant of
  nondegenerate  orthogonal  spaces  over  finite  fields  of odd order, up to
  isometry.  Thus, discriminants can be used to delineate the isometry type of
  an  orthogonal  form  in  even  (algebraic) dimension. The discriminant of a
  hermitian  form  is  not defined, and applying this operation on a hermitian
  form, will result in an error message.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> gram := InvariantQuadraticForm(GO(-1,4,5))!.matrix;;[0X
    [4Xgap> qform := QuadraticFormByMatrix(gram, GF(5));[0X
    [4X< quadratic form >[0X
    [4Xgap> DiscriminantOfForm( qform );[0X
    [4X"nonsquare"[0X
    [4X [0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X4.8 Recognition of sesquilinear forms preserved by a classical group[0X
  
  In  this  section,  we  describe  a function that was initially developed by
  Frank  Celler  (and which has now been adapted to [5XForms[0m) for the recognition
  of sesquilinear forms left invariant by a matrix group. More importantly, we
  should  stress  that  this  routine  differs  to that already offered by the
  [5XMeatAxe[0m  in  that  it  finds  sesquilinear  forms  preserved  up to [12Xscalars[0m.
  Eventually, the procedure used for finding preserved sesquilinear forms does
  use the [5XMeatAxe[0m but in some cases it can rule out the existence of preserved
  forms  without  calling  the [5XMeatAxe[0m. For more information on the algorithm,
  please see [CLNNPP08].
  
  [1X4.8-1 PreservedSesquilinearForms[0m
  
  [2X> PreservedSesquilinearForms( [0X[3Xgroup[0X[2X ) _____________________________[0Xoperation
  [6XReturns:[0X  a list of forms
  
  The  argument  [3Xgroup[0m  is a matrix group. The function uses random methods to
  find  all  of  the bilinear or unitary forms preserved by [3Xgroup[0m (the trivial
  form  is also a possibility) up to a scalar. Since the procedure relies on a
  pseudo-random  generator,  the  user  may need to execute the operation more
  than once to find all invariant sesquilinear forms.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> g := SU(4,3);[0X
    [4XSU(4,3)[0X
    [4Xgap> forms := PreservedSesquilinearForms(g);[0X
    [4X[ < hermitian form > ][0X
    [4Xgap> Display( forms[1] );[0X
    [4XHermitian form[0X
    [4XGram Matrix:[0X
    [4X . . . 2[0X
    [4X . . 2 .[0X
    [4X . 2 . .[0X
    [4X 2 . . .[0X
    [4X [0X
  [4X------------------------------------------------------------------[0X
  
  Here  is another example which shows that this procedure is suitable in some
  cases  where  using the [5XMeatAxe[0m is not applicable. Here, our matrix group is
  the  group  of  similarities  preserving  a  (hyperbolic)  bilinear  form on
  GF(3)^6.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> a := [ [ -1, 0, 0, -1, 0, 1 ], [ 0, -1, -1, 0, 0, 1 ], [0X
    [4X>        [ -1, 0, 0, 1, 0, 0 ],  [ 0, -1, 1, 0, 0, -1 ], [0X
    [4X>        [ 0, 0, 0, 0, 0, -1 ], [ 0, -1, -1, 1, 1, 1 ] ] * One(GF(3));;[0X
    [4Xgap> b := [ [ 1, -1, 1, -1, 1, -1 ], [ 1, 1, -1, 1, 1, 0 ], [0X
    [4X>        [ -1, 0, 1, 0, 0, 0 ], [ 0, -1, 0, 0, 0, 1 ], [0X
    [4X>        [ 1, 1, 1, 1, 1, 1 ], [ -1, 1, 1, 1, -1, 0 ] ] * One(GF(3));;[0X
    [4Xgap> g := Group( a, b );[0X
    [4X<matrix group with 2 generators>[0X
    [4Xgap> forms := PreservedSesquilinearForms( g );[0X
    [4X[ < bilinear form > ][0X
    [4Xgap> Display( forms[1] );[0X
    [4XBilinear form[0X
    [4XGram Matrix:[0X
    [4X . 1 . . . .[0X
    [4X 1 . . . . .[0X
    [4X . . . 1 . .[0X
    [4X . . 1 . . .[0X
    [4X . . . . . 1[0X
    [4X . . . . 1 .[0X
    [4Xgap> m := GModuleByMats( [a,b], GF(3) );;[0X
    [4Xgap> usemeataxe := MTX.InvariantBilinearForm(m);[0X
    [4Xfail[0X
    [4X [0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X4.9 The trivial form and some of its properties[0X
  
  It can be useful to work with trivial a quadratic or sesquilinear form, i.e.
  a  form  mapping  all  vectors, couples of vectors respectively, to the zero
  element  of  their  basefield. As mentioned in Section [14X4.1[0m, [5XForms[0m allows the
  construction of an object in the Category [10XIsTrivialForm[0m.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> mat := [[0,0,0],[0,0,0],[0,0,0]]*Z(7)^0;[0X
    [4X[ [ 0*Z(7), 0*Z(7), 0*Z(7) ], [ 0*Z(7), 0*Z(7), 0*Z(7) ], [0X
    [4X  [ 0*Z(7), 0*Z(7), 0*Z(7) ] ][0X
    [4Xgap> form1 := BilinearFormByMatrix(mat,GF(7));[0X
    [4X< trivial form >[0X
    [4Xgap> form2 := QuadraticFormByMatrix(mat,GF(7));[0X
    [4X< trivial form >[0X
    [4Xgap> form1 = form2;[0X
    [4Xtrue[0X
    [4Xgap> IsQuadraticForm(form1);[0X
    [4Xfalse[0X
    [4Xgap> IsSesquilinearForm(form1);[0X
    [4Xfalse[0X
    [4Xgap> mat := [[0,0],[0,0]]*Z(4)^0;[0X
    [4X[ [ 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2) ] ][0X
    [4Xgap> form3 := BilinearFormByMatrix(mat,GF(4));[0X
    [4X< trivial form >[0X
    [4Xgap> form3 = form1;[0X
    [4Xfalse[0X
    [4X [0X
  [4X------------------------------------------------------------------[0X
  
  As  we  have seen by the above example, there is only one trivial form for a
  given  vector  space over a finite field, and such a trivial form can result
  from  the  construction  of a quadratic form or a sesquilinear form, but the
  trivial  form  itself  is  none  of  these,  although  it  can  behave  as a
  sesquilinear or a quadratic form, depending on its arguments.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> mat := [[0,0,0,0],[0,0,0,0],[0,0,0,0],[0,0,0,0]]*Z(3)^0;[0X
    [4X[ [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], [0X
    [4X  [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ] ][0X
    [4Xgap> form := BilinearFormByMatrix(mat,GF(3));[0X
    [4X< trivial form >[0X
    [4Xgap> v := Random(GF(3)^4);[0X
    [4X[ Z(3), Z(3), 0*Z(3), Z(3) ][0X
    [4Xgap> [v,v]^form;[0X
    [4X0*Z(3)[0X
    [4Xgap> v^form;[0X
    [4X0*Z(3)[0X
    [4X [0X
  [4X------------------------------------------------------------------[0X
  
  The attributes and properties described in Section [14X4.7[0m are all applicable to
  trivial forms.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> mat := [[0,0,0],[0,0,0],[0,0,0]]*Z(11)^0;[0X
    [4X[ [ 0*Z(11), 0*Z(11), 0*Z(11) ], [ 0*Z(11), 0*Z(11), 0*Z(11) ], [0X
    [4X  [ 0*Z(11), 0*Z(11), 0*Z(11) ] ][0X
    [4Xgap> form := QuadraticFormByMatrix(mat,GF(121));[0X
    [4X< trivial form >[0X
    [4Xgap> IsReflexiveForm(form);[0X
    [4Xtrue[0X
    [4Xgap> IsAlternatingForm(form);[0X
    [4Xtrue[0X
    [4Xgap> IsSymmetricForm(form);[0X
    [4Xtrue[0X
    [4Xgap> IsOrthogonalForm(form);[0X
    [4Xfalse[0X
    [4Xgap> IsPseudoForm(form);[0X
    [4Xfalse[0X
    [4Xgap> IsSymplecticForm(form);[0X
    [4Xtrue[0X
    [4Xgap> IsDegenerateForm(form);[0X
    [4Xtrue[0X
    [4Xgap> IsSingularForm(form);[0X
    [4Xtrue[0X
    [4Xgap> BaseField(form);[0X
    [4XGF(11^2)[0X
    [4Xgap> GramMatrix(form);[0X
    [4X[ [ 0*Z(11), 0*Z(11), 0*Z(11) ], [ 0*Z(11), 0*Z(11), 0*Z(11) ], [0X
    [4X  [ 0*Z(11), 0*Z(11), 0*Z(11) ] ][0X
    [4Xgap> RadicalOfForm(form);[0X
    [4X<vector space over GF(11^2), with 3 generators>[0X
    [4X [0X
  [4X------------------------------------------------------------------[0X
  
