Stufe (algebra)
In field theory, the Stufe (/ʃtuːfə/; German: level) s(F) of a field 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. Albrecht Pfister 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 
.
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 forms, the product 
is itself a sum of squares, that is, 
for some 
. But since 
, we also have 
, and hence
and thus 
.
Positive characteristic
The Stufe 
for all fields 
with positive characteristic.
Proof: Let 
. It suffices to prove the claim for 
.
If 
then 
, so 
.
If 
consider the set 
of squares. 
is a subgroup of index 
in the cyclic group 
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 
.
*-algebra
In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings R and A, where R is commutative and A has the structure of an associative algebra over R. Involutive algebras generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation, matrices over the complex numbers and conjugate transpose, and linear operators over a Hilbert space and Hermitian adjoints.
Terminology
*-ring
In mathematics, a *-ring is a ring with a map * : A → A that is an antiautomorphism and an involution.
More precisely, * is required to satisfy the following properties:
for all x, y in A.
This is also called an involutive ring, involutory ring, and ring with involution. Note that the third axiom is actually redundant, because the second and fourth axioms imply 1* is also a multiplicative identity, and identities are unique.
Algebra
Algebra (from Arabic "al-jabr" meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Much early work in algebra, as the Arabic origin of its name suggests, was done in the Middle East, by mathematicians such as al-Khwārizmī (780 – 850) and Omar Khayyam (1048–1131).
†-algebra
A †-algebra (or, more explicitly, a †-closed algebra) is the name occasionally used in physics for a finite-dimensional C*-algebra. The dagger, †, is used in the name because physicists typically use the symbol to denote a hermitian adjoint, and are often not worried about the subtleties associated with an infinite number of dimensions. (Mathematicians usually use the asterisk, *, to denote the hermitian adjoint.) †-algebras feature prominently in quantum mechanics, and especially quantum information science.
References
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