linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...

, the Dieudonné determinant is a generalization of the

determinant of a matrix In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if an ...

to matrices over

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 us ...

s and

local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic num ...

s. It was introduced by . If ''K'' is a division ring, then the Dieudonné determinant is a

homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...

of groups from the group GL_''n''(''K'') of invertible ''n'' by ''n'' matrices over ''K'' onto the

abelianization In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal s ...

''K''^×/ 'K''^×, ''K''^×of the multiplicative group ''K''^× of ''K''. For example, the Dieudonné determinant for a 2-by-2 matrix is the residue class, in ''K''^×/ 'K''^×, ''K''^× of :

\det \left(\right) = 
\left\lbrace\right. .

Properties

Let ''R'' be a local ring. There is a determinant map from the matrix ring GL(''R'') to the abelianised unit group ''R''^×_ab with the following properties:Rosenberg (1994) p.64 * The determinant is invariant under

elementary row operation In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GL''n''(F) when F is a field. Left multiplication (pre-multip ...

s * The determinant of the identity is 1 * If a row is left multiplied by ''a'' in ''R''^× then the determinant is left multiplied by ''a'' * The determinant is multiplicative: det(''AB'') = det(''A'')det(''B'') * If two rows are exchanged, the determinant is multiplied by −1 * If R is commutative, then the determinant is invariant under transposition

Tannaka–Artin problem

Assume that ''K'' is finite over its centre ''F''. The

reduced norm In ring theory and related areas of mathematics a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' which is simple, and for which the center is exactly ''K''. (Note that ''not'' every simple a ...

gives a homomorphism ''N''_''n'' from GL_''n''(''K'') to ''F''^×. We also have a homomorphism from GL_''n''(''K'') to ''F''^× obtained by composing the Dieudonné determinant from GL_''n''(''K'') to ''K''^×/ 'K''^×, ''K''^×with the reduced norm ''N''₁ from GL₁(''K'') = ''K''^× to ''F''^× via the abelianization. The Tannaka–Artin problem is whether these two maps have the same kernel SL_''n''(''K''). This is true when ''F'' is locally compact but false in general.

References

* *.
Errata
* * {{DEFAULTSORT:Dieudonne determinant Linear algebra Determinants

Properties

Tannaka–Artin problem

See also

References