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stufe of a field
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(Definition)
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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 $2$ .
- 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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"stufe of a field" is owned by CWoo.
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Cross-references: power, finite, theorem, squares, sum, least number, field
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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 3411 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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Pending Errata and Addenda
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