PlanetMath: stufe of a field

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stufe of a field

(Definition)

The stufe of a field is the least number such that can be expressed as a sum of squares:

$\displaystyle -1=a_1^2+\cdots+a_n^2,$

where each $a_i\in F$ . If no such an

exists, then we say that the stufe of

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 , if can be expressed as a finite sum of squares, then the stufe of is a power of .

Bibliography

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

"stufe of a field" is owned by CWoo.

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Other names:

level of a field

Also defines:

stufe

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Cross-references: power, finite, squares, sum, least number, field
There are 2 references to this entry.

This is version 2 of stufe of a field, born on 2005-02-25, modified 2005-06-13.
Object id is 6829, canonical name is StufeOfAField.
Accessed 2374 times total.

Classification:

AMS MSC:	12D15 (Field theory and polynomials :: Real and complex fields :: Fields related with sums of squares )
	15A63 (Linear and multilinear algebra; matrix theory :: Quadratic and bilinear forms, inner products)

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