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Quaternion algebras and units
Mišlanová, Kristína ; Kala, Vítězslav (advisor) ; Růžička, Pavel (referee)
The aim of this work is to study the Hamiltonian quaternions H and quaternion alge- bras. The first two chapters are based on the article Quaternion algebras by K. Conrad and the rest on the book Quaternion algebras by J. Voight. In the beginning, we mainly develop the theory about quaternions, quaternion algebras and study equivalent condi- tions for being a split or non-split quaternion algebra. After that, we also characterize, up to isomorphism, quaternion algebras over several fields such as R, C or Fp. In the third chapter, the thesis deals with orders in quaternion algebras, especially Lipschitz and Hurwitz order. The fourth chapter is dedicated to the relationship between unit quaternions and rotations in R3 , thanks to which we can characterize finite subgroups of H1 , or equivalently H× . This result will be used in the last chapter, where we are mainly focused on the problem of characterization of the group of units in orders in Hamiltonian quaternions. 1
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