158 CHAPTER VII. 



Proceeding with this form as we did with f, we reach a form 



of determinant =j= 0. After in 1 such steps we reach the form 142). 



169. Certain of the a { in 142) are squares and the others are 

 not -squares in the GrF[p K ']. By applying a suitable substitution 

 which interchanges the /, we may suppose that in the resulting form 

 ,,...,, are squares, say a 2 , . . ., a?, while a,+i, . . ., a OT are not- 

 squares, say vaJLj_i, . . ., vttm, v being a particular not -square. Apply- 

 ing the substitution 



i; = r^ (<-!,...,) 



our form is transformed into 



i s+1 



Furthermore , we can transform f s into f s + 2 and vice versa. In 

 fact, the substitution of determinant a 2 + /3 2 



!$=&-/?!,, g} = /3g f +& 



transforms |?+ || into ( 2 -f /3 2 )(||-f ||). By 64, a and may be 

 chosen in the GrF[p n ~\, p > 2, such that 



We have therefore only two canonical forms, f m and /_i. The 

 latter form may be dropped if m be odd. Indeed, f m i can, for m 

 odd, be transformed into 



/=(!; + || + ... + !>,). 



But the linear group leaving /" invariant leaves also 



i =- & -4- & -4- - - 4- 2 



/m |_ i 63 r i 6m 



invariant. We may therefore state the theorem: 



The group of all linear homogeneous m-ary substitutions in the 

 GF\j> n ~\, p > 2, which leave invariant a quadratic form f belonging to 

 the field and of determinant not zero, can be transformed by a linear 

 homogeneous m-ary substitution belonging to the field into the group of 

 all linear homogeneous m-ary substitutions in the GrF\j> n ~] which leave 

 invariant 



tvhere ft = 1 for m odd, but p = 1 or a particular not -square v for m 

 even. 



