Sums of squares are studied since antiquity, starting from the determination of pythagorean triples of integers, Bramagupta’s composition formula for sums of two squares and Lagrange’s four squares theorem.
We focus on the study of the so-called pythagoras number of a commutative ring, defined as the smallest number p such that every sum of squares is equal to a sum of p squares.
We introduce and explore the sophisticated methods involved in the study of pythagoras numbers of fields. These will include tools from the algebraic theory of quadratic forms, in particular the Cassels-Pfister Theorem and upper bounds for the pythagoras numbers for certain extensions.
To obtain lower bounds for the pythagoras number of a field is generally an even more difficult problem. We aim in particular to exhibit the diverse methods and tools from algebraic geometry that apply for determining the lengths of sums of squares in function fields of real surfaces and of curves over number fields.