Hello,


Joel Dogde, in a comment on his question "Roots of unity in different completions of a number field", says the following, about the analogy between number fields and function fields : 


*Number of roots of unity in number fields is something like the size of the constant field for function fields.*


Could anyone explain that ?


Thanks.



Why the roots of unity are the analogs of constants ?
One answer: the roots of unity in $K$ are [precisely the elements](https://mathoverflow.net/questions/10911/english-reference-for-a-result-of-kronecker) of $K$ which have absolute value $1$ for every absolute value on $K$; the elements of the constant field have this property for function fields.


