In mathematics, the Pythagoras number or reduced height of a field describes the structure of the set of squares in the field. The Pythagoras number of a field is the smallest positive integer such that every sum of squares in is a sum of squares.
A Pythagorean field is a field with Pythagoras number 1: that is, every sum of squares is already a square.
Examples Every non-negative real number is a square, so . For a finite field of odd characteristic, not every element is a square, but all are the sum of two squares, so . By Lagrange's four-square theorem, every positive rational number is a sum of four squares, and not all are sums of three squares, so .
Properties Every positive integer occurs as the Pythagoras number of some formally real field.
The Pythagoras number is related to the Stufe by . If is not formally real then , and both cases are possible: for we have , whereas for we have , .
As a consequence, the Pythagoras number of a non-formally-real field is either a power of 2, or 1 more than a power of 2. All such cases occur: i.e., for each pair of the form or , there exists a field such that . For example, quadratically closed fields and fields of characteristic 2 give ; for primes , and the p-adic field give ; for primes , gives and gives ; gives , and the function field gives .
The Pythagoras number is related to the height of a field : if is formally real then is the smallest power of 2 which is not less than ; if is not formally real then .
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