# 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).
Title stufe of a field StufeOfAField 2013-03-22 15:06:01 2013-03-22 15:06:01 CWoo (3771) CWoo (3771) 5 CWoo (3771) Definition msc 15A63 msc 12D15 level of a field TheoremsOnSumsOfSquares stufe