Stufe (algebra)

In field theory, a branch of mathematics, the Stufe (//; German: level) "s"("F") of a field "F" is the least number of squares that sum to −1. If −1 cannot be written as a sum of squares, "s"("F") = formula_1. In this case, "F" is a formally real field. Albrecht Pfister proved that the Stufe, if finite, is always a power of 2, and that conversely every power of 2 occurs.
Powers of 2.
If formula_2 then formula_3 for some natural number formula_4.
"Proof:" Let formula_5 be chosen such that formula_6. Let formula_7. Then there are formula_8 elements formula_9 such that
Both formula_11 and formula_12 are sums of formula_13 squares, and formula_14, since otherwise formula_15, contrary to the assumption on formula_4.
According to the theory of Pfister forms, the product formula_17 is itself a sum of formula_13 squares, that is, formula_19 for some formula_20. But since formula_21, we also have formula_22, and hence
and thus formula_24.
Positive characteristic.
Any field formula_25 with positive characteristic has formula_26.
"Proof:" Let formula_27. It suffices to prove the claim for formula_28.
If formula_29 then formula_30, so formula_31.
If formula_32 consider the set formula_33 of squares. formula_34 is a subgroup of index formula_35 in the cyclic group formula_36 with formula_37 elements. Thus formula_38 contains exactly formula_39 elements, and so does formula_40.
Since formula_28 only has formula_42 elements in total, formula_38 and formula_40 cannot be disjoint, that is, there are formula_45 with formula_46 and thus formula_47.
Properties.
The Stufe "s"("F") is related to the Pythagoras number "p"("F") by "p"("F") ≤ "s"("F") + 1. If "F" is not formally real then "s"("F") ≤ "p"("F") ≤ "s"("F") + 1. The additive order of the form (1), and hence the exponent of the Witt group of "F" is equal to 2"s"("F").