Submission: 2008, Oct 6
We show as the main result, that the Pythagoras number of the function field of a conic is 2, only if the ground field is hereditarily pythagorean. Furthermore, we consider function fields of certain quadrics and prove some 'Going Down'-results for the finiteness of the Pythagoras number from the function field to the ground field (and all its finite extensions).
2000 Mathematics Subject Classification: 11E10, 12D15, 14H05, 12F10, 12J10, 14H05
Keywords and Phrases: Pythagoras number, Quadrics, Conics, Hereditarily Pythagorean Fields
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