Submission: 2004, Oct 5
It is shown that the Pythagoras number of a real function field of a hyperelliptic curve C with good reduction defined over the real formal power series field is equal to 2. A main tool in the proof is the explicit description of the Brauer group of C. If the function field of such a curve is non-real then it is shown that its Pythagoras number is 3.
2000 Mathematics Subject Classification: 11E10, 12D15, 12E15
Keywords and Phrases: Pythagoras number, Henselian discrete valued fields, Pythagorean fields, Brauer groups
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