stufe of a field

The stufe of a field $F$ is the least number $n$ such that $-1$ can be expressed as a sum of $n$ squares:

-1=a_{1}^{2}+\\cdots+a_{n}^{2},

where each $a_{i}\\in F$ . If no such an $n$ exists, then we say that the stufe of $F$ is $\\infty$ .

Remarks.

•
The word “stufe”, meaning “level” in German, is attributed to mathematician Albrecht Pfister.
•
A theorem of Pfister asserts that in a field $F$ , if $-1$ can be expressed as a finite sum of squares, then the stufe of $F$ is a power of $2$ .

References

1 A. Pfister, Zur Darstellung definiter Funktionen als Summe von Quadraten, Inventiones Mathematicae (1967).
2 A. R. Rajwade, Squares, Cambridge University Press (1993).

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