"Quaternion algebras and quadratic forms"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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==introduction== | ==introduction== | ||
| − | + | * consider an algebra 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 |
* division algebra | * division algebra | ||
* matrix algebra | * matrix algebra | ||
| − | * http://www.maths.tcd.ie/pub/ims/bull57/S5701.pdf | + | |
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| + | ==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. | ||
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| + | ==related items== | ||
| + | * [[Steinberg symbol]] | ||
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| + | ==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 | ||
* [http://uwspace.uwaterloo.ca/bitstream/10012/3656/1/second2.pdf Quaternion algebras and quadratic forms], Master's thesis, Zi Yang Sham, University of Waterloo | * [http://uwspace.uwaterloo.ca/bitstream/10012/3656/1/second2.pdf Quaternion algebras and quadratic forms], Master's thesis, Zi Yang Sham, University of Waterloo | ||
| − | * www.math.virginia.edu/~ww9c/kranec.pdf[[분류:개인노트 | + | * www.math.virginia.edu/~ww9c/kranec.pdf |
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| + | [[분류:개인노트]] | ||
[[분류:math and physics]] | [[분류:math and physics]] | ||
[[분류:math]] | [[분류:math]] | ||
2013년 11월 29일 (금) 08:45 판
introduction
- consider an algebra 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
- 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 \(i^2=a\), \(j^2=b\), \(ij=-ji=k\). In this case the algebra represents an element of order 2 in the Brauer group of 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.
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