Quartic rings can be explicitly parametrized

You need to know: Matrix, invertible matrix, determinant of a matrix, group, abelian group, group ${\mathbb Z}^n$ , ring, commutative ring, isomorphic groups and rings.

Background: Each commutative ring R is an abelian group with respect to addition. We say that R has rank n if this group is isomorphic to ${\mathbb Z}^n$ . Rings of rank $n=4$ are called quartic rings. An integral ternary quadratic form is an expression of the form $ax^4+bx^3y+cx^2y^2+dxy^3+ey^4$ with $a,b,c,d,e \in {\mathbb Z}$ . We say that two such forms A and B are linearly independent over ${\mathbb Q}$ if $uA+vB=0$ with rational $u,v$ is possible only if $u=v=0$ . Let $GL_n({\mathbb Z})$ be the set of invertible $n \times n$ matrices with integer entries, and let $GL_2^{\pm 1}({\mathbb Q})$ be the set of $2 \times 2$ matrices with rational entries and determinant $\pm 1$ .

The Theorem: On 29th February 2004, Manjul Bhargava submitted to the Annals of Mathematics a paper in which he proved the existence of a canonical bijection between isomorphism classes of nontrivial quartic rings and $GL_3({\mathbb Z}) \times GL^{\pm 1}_2({\mathbb Q})$ -equivalence classes of pairs $(A,B)$ of integral ternary quadratic forms where A and B are linearly independent over ${\mathbb Q}$ .

Short context: Explicit parametrizations for rings of ranks $n=2$ (known as quadratic rings) and $n=3$ (cubic rings) are relatively easy and well-known. The Theorem provides such a parametrization for quartic rings. In a later work, Bhargava also obtained explicit parametrization for rings of rank $n=5$ (quintic rings).