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:
$$1={a}_{1}^{2}+\mathrm{\cdots}+{a}_{n}^{2},$$ 
where each ${a}_{i}\in F$. If no such an $n$ exists, then we say that the stufe of $F$ is $\mathrm{\infty}$.
Remarks.

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The word “stufe”, meaning “level” in German, is attributed to mathematician Albrecht Pfister.

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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 

Canonical name  StufeOfAField 
Date of creation  20130322 15:06:01 
Last modified on  20130322 15:06:01 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  5 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 15A63 
Classification  msc 12D15 
Synonym  level of a field 
Related topic  TheoremsOnSumsOfSquares 
Defines  stufe 