Quaternion algebras and quadratic forms

introduction

• let $F$ be a field
• consider a quaternion algebra defined by $F[i,j]/(i^2=a,j^2=b,ij=-ji)$
• we denote it as

$$\left(\frac{a,b}{F}\right)$$

• 4 dimensional algebra over $F$ with basis $1,i,j,k$ and multiplication rules \(i^2=a\), \(j^2=b\), \(ij=-ji=k\).
• 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 $2\times 2$ matrices over $F$: the latter case is termed split

quaternion algebra as a quadratic space

• let us consider the algebra $A=\left(\frac{a,b}{F}\right)$
• we regard it as a quadratic space associated with the quadratic form $(1,-a,-b,ab)$

Hilbert symbol

• 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.

related items

• Steinberg symbol
• Quadratic forms over p-adic integer rings

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
• http://www.math.virginia.edu/~ww9c/kranec.pdf