"Quaternion algebras and quadratic forms"의 두 판 사이의 차이

2013년 11월 29일 (금) 08:51 판

let $F$ be a field
consider a quaternion algebra $(a,b)_F$ defined by $F[i,j]/(i^2=a,j^2=b,ij=-ji)$
it is an example of a central simple algebra (see Brauer group)
it is either a division algebra or isomorphic to the matrix algebra of 2×2 matrices over $F$: the latter case is termed split

The Hilbert symbol can also be used to denote the central simple algebra over $F$ with basis $1,i,j,k$ and multiplication rules $i^2=a$, $j^2=b$, $ij=-ji=k$. In this case the algebra represents an element of order 2 in the Brauer group of $F$, which is identified with -1 if it is a division algebra and +1 if it is isomorphic to the algebra of 2 by 2 matrices.

Lewis, David W. 2006. “Quaternion Algebras and the Algebraic Legacy of Hamilton’s Quaternions.” Irish Mathematical Society Bulletin (57): 41–64. http://www.maths.tcd.ie/pub/ims/bull57/S5701.pdf
Quaternion algebras and quadratic forms, Master's thesis, Zi Yang Sham, University of Waterloo
www.math.virginia.edu/~ww9c/kranec.pdf

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 ==introduction==
-* consider an algebra defined by $F[i,j]/(i^2=a,j^2=b,ij=-ji)$
+* let $F$ be a field
+* consider a quaternion algebra $(a,b)_F$ defined by $F[i,j]/(i^2=a,j^2=b,ij=-ji)$
 * it is an example of a central simple algebra (see [[Brauer group]])
-* classification of quaternion algebras over fields
+* it is either a division algebra or isomorphic to the matrix algebra of 2×2 matrices over $F$: the latter case is termed split
-* division algebra
-* matrix algebra
 ==Hilbert symbol==
-* The Hilbert symbol can also be used to denote the central simple algebra over ''K'' with basis 1,''i'',''j'',''k'' and multiplication rules <math>i^2=a</math>, <math>j^2=b</math>, <math>ij=-ji=k</math>.  In this case the algebra represents an element of order 2 in the [[Brauer group]] of ''K'', which is identified with -1 if it is a division algebra and +1 if it is isomorphic to the algebra of 2 by 2 matrices.
+* The Hilbert symbol can also be used to denote the central simple algebra over $F$ with basis $1,i,j,k$ and multiplication rules <math>i^2=a</math>, <math>j^2=b</math>, <math>ij=-ji=k</math>.  In this case the algebra represents an element of order 2 in the [[Brauer group]] of $F$, which is identified with -1 if it is a division algebra and +1 if it is isomorphic to the algebra of 2 by 2 matrices.