Abstract
The quadratic forms in three variables over the field are classified. Some remarks are made about the group of equivalences of the quadratic forms.
1. Introduction
By a quadratic form, we understand a homogeneous quadratic polynomial in variables where the 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 . 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 over associates to each quadratic form a symmetric matrix so that the quadratic form is . Under a change of variables, the matrix changes according to where is nonsingular. Such a change does not preserve the eigenvalues of . The only invariants are the signs of the eigenvalues; as such, every matrix may be reduced to a diagonal matrix in which every entry is , , or . The number of nonzero diagonal entries is the rank of the quadratic form; one can also sensibly define the signature of the quadratic form to be the difference between the number of positive and number of negative entries when 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 , the situation is different; that is, if the matrix is allowed to belong to rather than , 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 . Finally, if the matrix is orthogonal then the eigenvalues of 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 is 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 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 is . It is a group of order : 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 of the first two rows. It is easy to see that every element satisfies ; where is the identity matrix by using the Hamilton-Cayley theorem: in fact, there are only four inequivalent characteristic polynomials. A Sylow -group is generated by the matrix but the subgroup is not normal. Hence, by Sylow’s Theorem there must be Sylow -groups and elements of order .
The “Heisenberg” upper triangular unipotent matrices form a Sylow -subgroup of order as do the corresponding lower triangular unipotent matrices; hence, neither is normal and the number of Sylow -subgroups can be either , or . Each of these subgroups is not abelian and has at least two elements of order and therefore the Sylow -subgroups must each be isomorphic to the dihedral group of order . The sets of matrices of the form and comprise two more Sylow -subgroups and so or . In fact, it may be shown that . For many more details, we refer to [8] which is, in effect, an entire course about the simple group of order and is based almost on entirely on the Sylow Theorem.
As regards the Sylow -subgroups we must have , or . The group generated by generates a nonnormal subgroup of order three so . In fact it turns out that .
Actually, group is isomorphic to the unique simple group of order , the smallest order simple group after apart from the cyclic groups of prime order.
The orthogonal group over is the trivial group since if the first row of a matrix consists of ’s the other diagonal entries of will be zero.
3. Preliminary Reduction
There are, in principle, quadratic forms but some are equivalent by transformations from the symmetric group . There are forms that have no square terms so by symmetry such forms can be reduced toSimilarly there are forms that contain and they can be reduced to There are forms that contain one square and another that contain two squares and these forms may be reduced to respectively. We note further that in (3) the second and third forms and the sixth and seventh are equivalent, respectively, by interchanging and , whereas in (4) the second and third and fifth and sixth forms are equivalent, respectively, by interchanging and . As such we have now reduced the original list of forms to the following : .
4. Reduction to Canonical Form
Now we shall explain how the 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)(2)(3)(4)(5)(6)(7) canonical form(8)(9)(10)(11) canonical form(12)(13) canonical form(14)(15)(16)(17) canonical form(18) canonical form(19)(20).
5. The Canonical Forms
In this section, we shall justify the fact that the five canonical forms are mutually inequivalent. Clearly is only equivalent to itself. Next, under a general transformation can change only to and so it can only be equivalent to and , allowing for symmetry.
Next, we consider . Under a general nonsingular transformation is transformed to If were equivalent to then we would have and hence . Since the terms in and are absent we can only have but then the coefficient of must be zero and hence is not equivalent to .
Next we suppose that is equivalent to . This time we find that ; however, these conditions are self-contradictory since we would have that so that . Thus, is not equivalent to any of the other canonical forms.
Finally, we show that is not equivalent to . In fact, under (5) the form transforms to We consider a number of cases. Suppose first of all that . Then, , , and so that the transformation is singular. In the second case, suppose that . Then, we must have that and ; however, it then follows that and again the transformation is singular. The last possibility starting from the term in (7) is that and . Then, we must have that and and again the transformation is singular.
Theorem 1. Over every quadratic form in three variables is equivalent to precisely one of .
Competing Interests
The author hereby declares that there is no financial or other conflict of interests with regard to the dissemination of this paper.
Acknowledgments
The author thanks Akaki Tikaradze for helpful discussions.