Reduced Norm
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In
ring theory In algebra, ring theory is the study of rings— algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their re ...
and related areas 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 ...
a central simple algebra (CSA) over 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 ...
''K'' is a finite-dimensional associative ''K''-algebra ''A'' which is
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
, and for which the
center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentrici ...
is exactly ''K''. (Note that ''not'' every simple algebra is a central simple algebra over its center: for instance, if ''K'' is a field of characteristic 0, then the
Weyl algebra In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form : f_m(X) \partial_X^m + f_(X) \partial_X^ + \cdots + f_1(X) \partial_X + f_0(X). More prec ...
K ,\partial_X/math> is a simple algebra with center ''K'', but is ''not'' a central simple algebra over ''K'' as it has infinite dimension as a ''K''-module.) For example, the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s C form a CSA over themselves, but not over the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s R (the center of C is all of C, not just R). The
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s H form a 4-dimensional CSA over R, and in fact represent the only non-trivial element of the
Brauer group Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:- * Alfred Brauer (1894–1985), German-American mathematician, brother of Richard * Andreas Brauer (born 1973), German film producer * Arik ...
of the reals (see below). Given two central simple algebras ''A'' ~ ''M''(''n'',''S'') and ''B'' ~ ''M''(''m'',''T'') over the same field ''F'', ''A'' and ''B'' are called ''similar'' (or ''
Brauer equivalent Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:- * Alfred Brauer (1894–1985), German-American mathematician, brother of Richard * Andreas Brauer (born 1973), German film producer * Arik Br ...
'') if their division rings ''S'' and ''T'' are isomorphic. The set of all
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es of central simple algebras over a given field ''F'', under this equivalence relation, can be equipped with a
group operation In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. Thes ...
given by the
tensor product of algebras In mathematics, the tensor product of two algebras over a commutative ring ''R'' is also an ''R''-algebra. This gives the tensor product of algebras. When the ring is a field, the most common application of such products is to describe the prod ...
. The resulting group is called the
Brauer group Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:- * Alfred Brauer (1894–1985), German-American mathematician, brother of Richard * Andreas Brauer (born 1973), German film producer * Arik ...
Br(''F'') of the field ''F''.Lorenz (2008) p.159 It is always a
torsion group In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which every element has finite order. The exponent of such a group, if it exists, is the least common multiple of the orders of the elements. For examp ...
.Lorenz (2008) p.194


Properties

* According to the Artin–Wedderburn theorem a finite-dimensional simple algebra ''A'' is isomorphic to the matrix algebra ''M''(''n'',''S'') for some
division ring In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
''S''. Hence, there is a unique division algebra in each Brauer equivalence class.Lorenz (2008) p.160 * Every automorphism of a central simple algebra is an
inner automorphism In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group it ...
(this follows from the
Skolem–Noether theorem In ring theory, a branch of mathematics, the Skolem–Noether theorem characterizes the automorphisms of simple rings. It is a fundamental result in the theory of central simple algebras. The theorem was first published by Thoralf Skolem in 1927 in ...
). * The
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of a central simple algebra as a vector space over its centre is always a square: the degree is the square root of this dimension.Gille & Szamuely (2006) p.21 The Schur index of a central simple algebra is the degree of the equivalent division algebra:Lorenz (2008) p.163 it depends only on the
Brauer class Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:- * Alfred Brauer (1894–1985), German-American mathematician, brother of Richard * Andreas Brauer (born 1973), German film producer * Arik Br ...
of the algebra.Gille & Szamuely (2006) p.100 * The period or exponent of a central simple algebra is the order of its Brauer class as an element of the Brauer group. It is a divisor of the index,Jacobson (1996) p.60 and the two numbers are composed of the same prime factors.Jacobson (1996) p.61Gille & Szamuely (2006) p.104 * If ''S'' is a simple
subalgebra In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations. "Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operat ...
of a central simple algebra ''A'' then dim''F'' ''S'' divides dim''F'' ''A''. * Every 4-dimensional central simple algebra over a field ''F'' is isomorphic to a
quaternion algebra In mathematics, a quaternion algebra over a field ''F'' is a central simple algebra ''A'' over ''F''See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2. that has dimension 4 over ''F''. Every quaternion algebra becomes a ma ...
; in fact, it is either a two-by-two
matrix algebra In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ''U ...
, or a
division algebra In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a fie ...
. * If ''D'' is a central division algebra over ''K'' for which the index has prime factorisation ::\mathrm(D) = \prod_^r p_i^ \ :then ''D'' has a tensor product decomposition ::D = \bigotimes_^r D_i \ :where each component ''D''''i'' is a central division algebra of index p_i^, and the components are uniquely determined up to isomorphism.Gille & Szamuely (2006) p.105


Splitting field

We call a field ''E'' a ''splitting field'' for ''A'' over ''K'' if ''A''⊗''E'' is isomorphic to a matrix ring over ''E''. Every finite dimensional CSA has a splitting field: indeed, in the case when ''A'' is a division algebra, then a
maximal subfield In algebra, a subfield of an algebra ''A'' over a field ''F'' is an ''F''-subalgebra that is also a field. A maximal subfield is a subfield that is not contained in a strictly larger subfield of ''A''. If ''A'' is a finite-dimensional central simp ...
of ''A'' is a splitting field. In general by theorems of Wedderburn and Koethe there is a splitting field which is a
separable extension In field theory, a branch of algebra, an algebraic field extension E/F is called a separable extension if for every \alpha\in E, the minimal polynomial of \alpha over is a separable polynomial (i.e., its formal derivative is not the zero polyno ...
of ''K'' of degree equal to the index of ''A'', and this splitting field is isomorphic to a subfield of ''A''.Jacobson (1996) pp.27-28Gille & Szamuely (2006) p.101 As an example, the field C splits the quaternion algebra H over R with : t + x \mathbf + y \mathbf + z \mathbf \leftrightarrow \left(\right) . We can use the existence of the splitting field to define reduced norm and reduced trace for a CSA ''A''.Gille & Szamuely (2006) pp.37-38 Map ''A'' to a matrix ring over a splitting field and define the reduced norm and trace to be the composite of this map with determinant and trace respectively. For example, in the quaternion algebra H, the splitting above shows that the element ''t'' + ''x'' i + ''y'' j + ''z'' k has reduced norm ''t''2 + ''x''2 + ''y''2 + ''z''2 and reduced trace 2''t''. The reduced norm is multiplicative and the reduced trace is additive. An element ''a'' of ''A'' is invertible if and only if its reduced norm in non-zero: hence a CSA is a division algebra if and only if the reduced norm is non-zero on the non-zero elements.Gille & Szamuely (2006) p.38


Generalization

CSAs over a field ''K'' are a non-commutative analog to
extension field In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
s over ''K'' – in both cases, they have no non-trivial 2-sided ideals, and have a distinguished field in their center, though a CSA can be non-commutative and need not have inverses (need not be a
division algebra In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a fie ...
). This is of particular interest in noncommutative number theory as generalizations of
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 ...
s (extensions of the rationals Q); see noncommutative number field.


See also

*
Azumaya algebra In mathematics, an Azumaya algebra is a generalization of central simple algebras to ''R''-algebras where ''R'' need not be a field. Such a notion was introduced in a 1951 paper of Goro Azumaya, for the case where ''R'' is a commutative local rin ...
, generalization of CSAs where the base field is replaced by a commutative local ring *
Severi–Brauer variety In mathematics, a Severi–Brauer variety over a field ''K'' is an algebraic variety ''V'' which becomes isomorphic to a projective space over an algebraic closure of ''K''. The varieties are associated to central simple algebras in such a way ...
*
Posner's theorem In algebra, Posner's theorem states that given a prime polynomial identity algebra ''A'' with center ''Z'', the ring A \otimes_Z Z_ is a central simple algebra over Z_, the field of fractions In abstract algebra, the field of fractions of an in ...


References

* * * *


Further reading

* * {{cite book , last1=Gille , first1=Philippe , last2=Szamuely , first2=Tamás , title=Central simple algebras and Galois cohomology , series=Cambridge Studies in Advanced Mathematics , volume=101 , location=Cambridge , publisher=
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ...
, year=2006 , isbn=0-521-86103-9 , zbl=1137.12001 Algebras Ring theory