Quadratic Forms Beyond Arithmetic

Summary

Quadratic Forms Beyond ArithmeticAlexander MerkurjevRaman ParimalaCommunicated by Notices Associate Editor Han-Bom Moon 1. Introduction The concept of quadratic forms can be traced back to ancient civilizations such as the Babylonians and Greeks. The Greeks, particularly Euclid in his famous work Elements, presented geometric methods for solving quadratic equations. The Greeks’ focus on geometry and their methods continued to influence mathematicians for centuries. Rules for quadratic equations were also discussed in The Nine Chapters on the Mathematical Art, composed in China by 200 BCE. The general formula for solving a quadratic equation in one variable—equivalent to the modern symbolic formula—was first stated by the Indian mathematician Brahmagupta in his treatise Brāhmasphutasiddhānta in 628 CE. A quadratic form over a commutative ring $R$ is a homogeneous polynomial $\operatorname *{\textstyle \sum }a_{ij}x_i x_j$ of degree $2$ in $n$ variables $x_1,\ldots , x_n$ with coefficients $a_{ij}$ in $R$. In particular, the sum of squares $x_1^2+x_2^2+\ldots + x_n^2$ is a quadratic form defined over any $R$. The problem of representing integers as sums of squares dates back to ancient times. The Greeks, especially the Pythagoreans, were interested in the properties of numbers and their geometric interpretations. The concept of sums of squares is closely related to the Pythagorean theorem, conceived in Mesopotamia (1800 BC), first stated precisely in the Shulbha Sutra of Baudhayana (800 BC) and a statement of proof from China. In the seventh century, the Indian mathematician Brahmagupta considered what is now called Pell’s equation, $x^2-ay^2=1$, and found a method for its solution. One of the earliest and most significant results in the area of quadratic forms is Fermat’s theorem on sums of two squares. In the seventeenth century, Fermat stated that an odd prime number $p$ can be expressed as a sum of two squares if and only if $p$ is congruent to $1$ modulo $4$. Ano...