The problem of writing a totally positive element as a sum of squares has a long history in mathematics, going back to Bachet and Lagrange. While for some specific rings (like integers or polynomials over the rationals), there are known methods for decomposing an element into a sum of squares, in general, for many other important rings and fields, the problem is still widely open. In this paper, we present an explicit algorithm for decomposing an element of an arbitrary global field (either a number field or a global function field) into a sum of squares of minimal length.

翻译：写一个完全正面的元素作为方块之和的问题在数学上有着悠久的历史,可以追溯到Bachet和Lagrange。对于某些特定的环形(如整数或对理性的多数值)来说,有已知的方法可以将一个元素分解成一个方块的总和,一般来说,对于许多其他重要的环形和字段来说,这个问题仍然广泛存在。在本文中,我们提出了一个明确的算法,将一个任意的全球域(无论是数字字段还是全球函数字段)的元素分解成一个最小长度的正方块。