# Classifying Quadratic Forms OverZ2 in Three Variables

Authors: Gerard Thompson
Journal: Journal of Mathematics
 (2016)

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## Abstract

The quadratic forms in three variables over the fieldZ2 are classified. Some remarks are made about the group of equivalences GL(3,Z2) of the quadratic forms.


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## Body

## 1. Introduction

By aquadratic form, we understand a homogeneous quadratic polynomial in n variables ∑i,jaijxixj where the aij belong to a field or at least a commutative ring. In this article, we shall consider the equivalence of quadratic forms in three variables over the field Z2. As our references suggest, the study of quadratic forms over finite fields lies at the nexus of several areas of mathematics, combinatorics, cryptography, and the theory of algebraic curves, to name but three of them.The standard approach to classifying quadratic forms overR associates to each quadratic form a symmetric matrix A so that the quadratic form is xtAx. Under a change of variables, the matrix A changes according to PtAP where P is nonsingular. Such a change does not preserve the eigenvalues of A. The only invariants are thesigns of the eigenvalues; as such, every matrix A may be reduced to a diagonal matrix in which every entry is 1, -1, or 0. The number of nonzero diagonal entries is therank of the quadratic form; one can also sensibly define thesignature of the quadratic form to be the difference between the number of positive and number of negative entries when A has been diagonalized. Conventions vary in these definitions. Another approach is simply to repeatedly “complete the square” so as to reduce the quadratic form to diagonal form. Over C, the situation is different; that is, if the matrix P is allowed to belong to GL(n,C) rather than GL(n,R), the distinction between positive and negative eigenvalues disappears and a quadratic form may always be reduced to diagonal form in which every nonzero entry is +1. Finally, if the matrix P is orthogonal then the eigenvalues of A are preserved and one obtains the finite-dimensional spectral theorem: for further details see [1].It is not possible to associate a symmetric matrix to a quadratic form when the field isZ2 since the cross terms would be all be zero. Instead, one could work simply with an upper triangular matrix. For background material about quadratic forms, we refer to [1, 2] and a much more recent account in [3]. For more theoretical considerations, we refer to [4] and references therein. In [5], the radical (maximal isotropic subspace) of a certain class of quadratic forms over fields of characteristic 2 is determined. In [6], among other things, the author studies the zeros of a quadratic form. Yet another direction [7] concerns pencils of quadratic forms over finite fields.


## 2. Group of Equivalences

The group of equivalences under which a quadratic form can be changed isGL(3,Z2). It is a group of order 168: the first row is arbitrary except that it must not consist of all zeros giving seven choices; there are six possibilities for the second row and four for the third since it cannot be a sum modulo 2 of the first two rows. It is easy to see that every element g∈GL(3,Z2) satisfies g7=I; where I is the 3×3 identity matrix by using the Hamilton-Cayley theorem: in fact, there are only four inequivalent characteristic polynomials. A Sylow 7-group is generated by the matrixT=111101100 but the subgroup is not normal. Hence, by Sylow’s Theorem there must be 8 Sylow 7-groups and 48 elements of order 7.The “Heisenberg” upper triangular unipotent matrices form a Sylow2-subgroup of order 8 as do the corresponding lower triangular unipotent matrices; hence, neither is normal and the number of Sylow 2-subgroups k2 can be either 3,7, or 21. Each of these subgroups is not abelian and has at least two elements of order 2 and therefore the Sylow 2-subgroups must each be isomorphic to the dihedral group D8 of order 8. The sets of matrices of the form 100x1yz01 and 1xz0100y1 comprise two more Sylow 2-subgroups and so k2=7 or 21. In fact, it may be shown that k2=21. For many more details, we refer to [8] which is, in effect, an entire course about the simple group of order 168 and is based almost on entirely on the Sylow Theorem.As regards the Sylow3-subgroups we must have k3=1,4,7, or 28. The group generated by 001100010 generates a nonnormal subgroup of order three so k3≠1. In fact it turns out that k3=28.Actually, groupGL(3,Z2) is isomorphic to the uniquesimple group of order 168, the smallest order simple group after A5 apart from the cyclic groups of prime order.The3×3 orthogonal group over Z2 is the trivial group since if the first row of a matrix P consists of 1’s the other diagonal entries of PPt will be zero.


## 3. Preliminary Reduction

There are, in principle,64 quadratic forms but some are equivalent by transformations from the symmetric group S3. There are 8 forms that have no square terms so by symmetry such forms can be reduced to(1)xy,xy+xz,xy+xz+zx.Similarly there are 8 forms that contain x2+y2+z2 and they can be reduced to (2)x2+y2+z2,x2+y2+z2+xy,x2+y2+z2+xy+xz,x2+y2+z2+xy+xz+zx.There are 24 forms that contain one square and another 24 that contain two squares and these forms may be reduced to (3)x2,x2+xy,x2+xz,x2+yz,x2+xy+xz,x2+xy+yz,x2+xz+yz,x2+xy+yz+zx,(4)x2+y2,x2+y2+xz,x2+y2+yz,x2+y2+xy,x2+y2+xy+xz,x2+y2+xy+yz,x2+y2+xz+yz,x2+y2+xy+yz+zx,respectively. We note further that in (3) the second and third forms and the sixth and seventh are equivalent, respectively, by interchanging y and z, whereas in (4) the second and third and fifth and sixth forms are equivalent, respectively, by interchanging x and y. As such we have now reduced the original list of 64 forms to the following 20: x2+y2+z2,x2+y2+z2+xy,x2+y2+z2+xy+xz,x2+y2+z2+xy+xz+zxx2+y2,x2+y2+xz,x2+y2+xy,x2+y2+xy+xz,x2+y2+xz+yz,x2+y2+xy+yz+zxx2,x2+xy,x2+yz,x2+xy+xz,x2+xy+yz,x2+xy+yz+zx0,xy,xy+xz,xy+xz+zx.


## 4. Reduction to Canonical Form

Now we shall explain how the20 forms given at the end of the previous section may be further reduced to just five canonical forms. We shall denote equivalence of forms by ~ and in each case we given the accompanying transformation unless it is merely a permutation:(1)
x2+y2+z2~x2(x=X+Y,y=Y+Z,z=Z)(2)
x2+y2+z2+xy~xy+z2(x=X,y=X+Y,z=X+Y+Z)~x2+yz(3)
x2+y2+z2+xy+xz~x2+xz+z2(x=X,y=X+Y,z=X+Y+Z)~x2+xy+y2(4)
x2+y2+z2+xy+xy+xz~y2+yz+z2(x=X,y=X+Y,z=X+Y+Z)~x2+xy+y2(5)
x2+y2~x2(x=X+Y+Z,y=Y+Z,z=Z)(6)
x2+y2+xz~y2+xz(x=X,y=X+Y,z=Z)z~x2+yz(7)
x2+y2+xy canonical form(8)
x2+y2+xy+xz~y2+xz(x=Z,y=Y+Z,z=X+Y+Z)~x2+yz(9)
x2+y2+xz+yz~xy(x=Z,y=Y+Z,z=X+Y)(10)
x2+y2+xy+yz+zx~xy+z2(x=Z,y=Y+Z,z=X+Y+Z)(11)
x2 canonical form(12)
x2+xy~yz(x=Z,y=Y+Z,z=X+Y+Z)~xy(13)
x2+yz canonical form(14)
x2+xy+xz~xz(x=Z,y=Y+Z,z=X+Y)~xy(15)
x2+xy+yz~x2+yz(x=X+Z,y=Y+Z,z=X)(16)
x2+xy+yz+zx~xy(x=Z,y=Y+Z,z=X+Z)(17)
0 canonical form(18)
xy canonical form(19)
xy+xz~xy(x=Y,y=Y+Z,z=X+Y+Z)(20)
xy+xz+zx~xy+z2(x=X+Z,y=Y+Z,z=Z)~x2+yz.


## 5. The Canonical Forms

In this section, we shall justify the fact that the five canonical forms are mutually inequivalent. Clearly0 is only equivalent to itself. Next, under a general transformation x2 can change only to ax2+by2+cz2 and so it can only be equivalent to x2+y2 and x2+y2+z2, allowing for symmetry.Next, we considerxy. Under a general nonsingular transformation (5)x=aX+bY+cZ,y=dX+eY+fZ,z=gX+hY+iZ,xy is transformed to (6)adX2+bd+aeXY+cd+afZX+beY2+bf+ceZY+cfZ2.If xy were equivalent to x2+yz then we would have ad=1 and hence a=d=1. Since the terms in zx and z2 are absent we can only have c=f=0 but then the coefficient of yz must be zero and hence xy is not equivalent to x2+yz.Next we suppose thatxy is equivalent to x2+xy+y2. This time we find that ad=bd+ae=be=1; however, these conditions are self-contradictory since we would have that a=b=d=e=1 so that bd+ae=0. Thus, xy is not equivalent to any of the other canonical forms.Finally, we show thatx2+yz is not equivalent to x2+xy+y2. In fact, under (5) the form x2+yz transforms to (7)a2+dgX2+eg+dhXY+fg+diZX+eh+b2Y2+fh+eiZY+c2+fiZ2.We consider a number of cases. Suppose first of all that b=e=1. Then, h=0,g=1, i=0, and c=f=0 so that the transformation is singular. In the second case, suppose that b=h=1. Then, we must have that c=e=f=0 and d=1; however, it then follows that i=0 and again the transformation is singular. The last possibility starting from the Y2 term in (7) is that b=0 and e=h=1. Then, we must have that e=g=1 and c=f=i=0 and again the transformation is singular.Theorem 1.
OverZ2 every quadratic form in three variables is equivalent to precisely one of 0,xy,x2,x2+yz,x2+y2+xy.



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*Source: 7407638-2016-12-13.xml*