Remark 5.4.1.
In the function field setting, we have a much more straightforward alternative to the use of cyclotomic extensions: we may take the map to the Galois group of the base finite field. The point is that in this case we have an ample supply of everywhere unramified extensions of the base field (without quotation marks).
In the number field setting, using cyclotomic extensions as a proxy for abelian, everywhere unramified extensions is a rather productive idea even outside of class field theory. For one, it is the central premise of Iwasawa theory, in which one studies the behavior of class fields in certain towers of number fields and their relationship with -adic -functions (and other related concepts). For another, it is the starting point of -adic Hodge theory, in which one studies the relationship between different cohomology theories associated to algebraic varieties over local fields.