In
field theory, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Stufe (/
ʃtuːfə/; German: level) ''s''(''F'') of a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''F'' is the least number of squares that sum to −1. If −1 cannot be written as a sum of squares, ''s''(''F'') =
. In this case, ''F'' is a
formally real field
In mathematics, in particular in field theory and real algebra, a formally real field is a field that can be equipped with a (not necessarily unique) ordering that makes it an ordered field.
Alternative definitions
The definition given above i ...
.
Albrecht Pfister
Albrecht Pfister (c. 1420 – c. 1466) was one of the first European printers to use movable type, following its invention by Johannes Gutenberg. Working in Bamberg, Germany, he is believed to have been responsible for two innovations in the use ...
proved that the Stufe, if finite, is always a power of 2, and that conversely every power of 2 occurs.
[
]
Powers of 2
If then for some natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
.[Rajwade (1993) p.13][Lam (2005) p.379]
''Proof:'' Let be chosen such that . Let . Then there are elements such that
:
Both and are sums of squares, and , since otherwise , contrary to the assumption on .
According to the theory of Pfister form In mathematics, a Pfister form is a particular kind of quadratic form, introduced by Albrecht Pfister in 1965. In what follows, quadratic forms are considered over a field ''F'' of characteristic not 2. For a natural number ''n'', an ''n''-fold P ...
s, the product is itself a sum of squares, that is, for some . But since , we also have , and hence
:
and thus .
Positive characteristic
Any field with positive characteristic has .[Rajwade (1993) p.33]
''Proof:'' Let . It suffices to prove the claim for .
If then , so .
If consider the set of squares. is a subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
in the cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
with elements. Thus contains exactly elements, and so does .
Since only has elements in total, and cannot be disjoint, that is, there are with and thus .
Properties
The Stufe ''s''(''F'') is related to the Pythagoras number ''p''(''F'') by ''p''(''F'') ≤ ''s''(''F'') + 1.[Rajwade (1993) p.44] If ''F'' is not formally real then ''s''(''F'') ≤ ''p''(''F'') ≤ ''s''(''F'') + 1.[Rajwade (1993) p.228][Lam (2005) p.395] The additive order of the form (1), and hence the exponent of the Witt group
In mathematics, a Witt group of a field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear forms over the field.
Definition
Fix a field ''k'' of characteristic not equal to two. All vector spaces ...
of ''F'' is equal to 2''s''(''F'').[Milnor & Husemoller (1973) p.75][Lam (2005) p.380]
Examples
* The Stufe of a quadratically closed field is 1.[
* The Stufe of an algebraic number field is ∞, 1, 2 or 4 (Siegel's theorem).][ Examples are Q, Q(√−1), Q(√−2) and Q(√−7).][
* The Stufe of a ]finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
GF(''q'') is 1 if ''q'' ≡ 1 mod 4 and 2 if ''q'' ≡ 3 mod 4.
* The Stufe of a local field
In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact ...
of odd residue characteristic is equal to that of its residue field. The Stufe of the 2-adic field Q2 is 4.[Lam (2005) p.381]
Notes
References
*
*
*
Further reading
* {{cite book , last1=Knebusch , first1=Manfred , last2=Scharlau , first2=Winfried , title=Algebraic theory of quadratic forms. Generic methods and Pfister forms , others=Notes taken by Heisook Lee , series=DMV Seminar , volume=1 , location=Boston - Basel - Stuttgart , publisher=Birkhäuser Verlag , year=1980 , isbn=3-7643-1206-8 , zbl=0439.10011
Field (mathematics)