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'') =

\infty

. 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 is ...

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 u ...

proved that the Stufe, if finite, is always a power of 2, and that conversely every power of 2 occurs.

Powers of 2

s(F)\ne\infty

then

s(F)=2^k

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 ...

k

.Rajwade (1993) p.13Lam (2005) p.379 ''Proof:'' Let

k \in \mathbb N

be chosen such that

2^k \leq s(F) < 2^

. Let

n = 2^k

. Then there are

s = s(F)

elements

e_1, \ldots, e_s \in F\setminus\

such that :

0 = \underbrace_ + \underbrace_\;.

Both

a

and

b

are sums of

n

squares, and

a \ne 0

, since otherwise

s(F)< 2^k

, contrary to the assumption on

k

. 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 Pf ...

s, the product

ab

is itself a sum of

n

squares, that is,

ab = c_1^2 + \cdots + c_n^2

for some

c_i \in F

. But since

a+b=0

, we also have

-a^2 = ab

, and hence :

-1 = \frac = \left(\frac \right)^2 + \cdots + \left(\frac \right)^2,

and thus

s(F) = n = 2^k

Positive characteristic

Any field

F

with positive characteristic has

s(F) \le 2

.Rajwade (1993) p.33 ''Proof:'' Let

p = \operatorname(F)

. It suffices to prove the claim for

\mathbb F_p

. If

p = 2

then

-1 = 1 = 1^2

, so

s(F)=1

. If

p>2

consider the set

S=\

of squares.

S\setminus\

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 ...

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 ...

2

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 ...

\mathbb F_p^\times

with

p-1

elements. Thus

S

contains exactly

\tfrac2

elements, and so does

-1-S

. Since

\mathbb F_p

only has

p

elements in total,

S

and

-1-S

cannot be disjoint, that is, there are

x,y\in\mathbb F_p

with

S\ni x^2=-1-y^2\in-1-S

and thus

-1=x^2+y^2

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.228Lam (2005) p.395 The additive order of the form (1), and hence the

exponent Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...

of the

Witt group In mathematics, a Witt group of a field (mathematics), field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear form, symmetric bilinear forms over the field. Definition Fix a field ''k'' of characte ...

of ''F'' is equal to 2''s''(''F'').Milnor & Husemoller (1973) p.75Lam (2005) p.380

Examples

* The Stufe of a quadratically closed field is 1. * The Stufe of an

algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...

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 t ...

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