This paper presents algorithms for computing the length of a sum of squares and a Pythagoras element in a global field K of characteristic different from 2. In the first part of the paper, we present algorithms for computing the length in a non-dyadic and dyadic (if K is a number field) completion of K. These two algorithms serve as subsidiary steps for computing lengths in global fields. In the second part of the paper we present a procedure for constructing an element whose length equals the Pythagoras number of a global field, termed a Pythagoras element.
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We present a generalization of a polynomial factorization algorithm that works with ideals in maximal orders of global function fields. The method presented in this paper is intrinsic in the sense that it does not depend on the embedding of the ring of polynomials into the Dedekind domain in question.
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