Thyroid appearance of cells is viewed to determine whether they resemble normal or cancerous cells. There can be many nodules, which is termed a multinodular goitre, and this can sometimes be a toxic multinodular goitre. An enlarged thyroid gland is called a goitre. Goitres are present in some form in about 5% of people, and are the result of a large number of causes, including iodine deficiency, autoimmune disease (both Grave's disease and Hashimoto's thyroiditis), infection, inflammation, and infltrative disease such as sarcoidosis and amyloidosis. Sometimes no cause can be found, a state called "simple goitre". Some forms of goitre are
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