Quaternion algebras and quadratic forms
imported>Pythagoras0님의 2013년 11월 29일 (금) 08:51 판
introduction
- 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
Hilbert symbol
- 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.
expositions
- 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