Goitre in iodine in the soil. For example, in the English Midlands, the condition was known as Derbyshire Neck. In the United States, goitre was found in the Great Lakes, Midwest, and Intermountain regions. The condition now is practically absent in affluent nations, where table salt is supplemented with iodine. However, it is still prevalent in India, China, Central Asia, and Central Africa. Goitre had been prevalent in the alpine countries for a long time. Switzerland reduced the condition by introducing iodised salt in 1922. The Bavarian tracht in the Miesbach and Salzburg regions, which appeared in the 19th century, includes
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