Detailed example of a skew field different from Hamilton quaternion Do you have a reference of a detailed construction of a skew field different from the quaternions from Hamilton? I would appreciate if that would be accessible from the Internet.

I guess you're intending to ask for a skew-field (aka division ring) that is definitely not commutative.
Since $\Bbb H[x]$ is a right- (and left-) Ore domain, you can form a division ring of fractions for it nearly as you would for a commutative domain. Since it has infinite $\Bbb R$ dimension, it's clearly not the same thing as $\Bbb H$, and it contains a copy of $\Bbb H$ so it is not commutative.
It's the same basic idea: fractions of polynomials from $\Bbb H [x]$ where the denominator is a nonzero polynomial. The main difference is that the equivalence classes of fractions has to be more carefully defined. It's discussed in detail at wikipedia and also in Lam's Lectures on modules and rings where Ore domains are discussed.
