The construction of central simple division algebras over a number field is a pretty thing. I won't bore you with the details, but it is related to the Galois cohomology of the field in the center of the algebras.
Addition is done in the natural way.
In plain speech this means we use the presentation of K as a three-dimensional vector space over Q spanned by alpha, beta, gamma, which are three conjugates in K under the action of the Galois group. This is what the Brauer group is. I'm not sure I believe this. Hello, please let me ask some questions about galois-fields, because I have only a few information about this field. I'm not sure I believe this.