All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
Studying orthodontics in itself is permitted; however, it is not permissible for whoever studied this to carry out any forbidden correction, like splitting the teeth [in order to look better] without a need, but if it is done for a need because the teeth are deformed [or broken] then there is no harm to remove the deformation. The kind of splitting or teeth correction that is forbidden is the one that is done for the purpose of beautification and not the one that is done to repair a deformation [or a broken tooth] so that one would avoid what others would find as disgraceful in him.
Therefore, there is no harm for you to study orthodontics unless you know that you will be obliged during your studies to carry out forbidden beautification operations in which case it is not permissible.
Allaah Knows best.