Goitre an underactive thyroid, thyroid supplements are used as treatment. Sometimes a partial or complete thyroidectomy is required. Goitre is more common among women, but this includes the many types of goitre caused by autoimmune problems, and not only those caused by simple lack of iodine. Chinese physicians of the Tang Dynasty (618–907) were the first to successfully treat patients with goitre by using the iodine-rich thyroid gland of animals such as sheep and pigs—in raw, pill, or powdered form. This was outlined in Zhen Quan's (d. 643 AD) book, as well as several others. One Chinese book, "The Pharmacopoeia of
Trace field of a representation of formula_4 (see for example in ). The "invariant trace field" is equal to the trace field of the subgroup formula_7 generated by all squares of elements of formula_4 (a finite-index subgroup of formula_4). The invariant trace field of Fuchsian groups is stable under taking commensurable groups. This is not the case for the trace field; in particular the trace field is in general different from the invariant trace field. Let formula_4 be a Fuchsian group and formula_11 its trace field. Let formula_12 be the formula_11-subalgebra of the matrix algebra formula_14 generated by the preimages of elements of formula_4. The