Logical difficulties with cyclotomic fields
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The irreducibility of the general cyclotomic polynomials goes back to a lot of 19th century work starting with Gauss -- but unfortunately there are serious logical problems, arising from the temptation to embed cyclotomic fields, generated by so called "roots of unity", into the complex number field. While this has a pleasing geometric aspect, it also distorts what is really going on, and takes us outside the purely algebraic nature of the subject, with a dependence on analysis to create complex numbers and buttress the Fundamental Theorem of Algebra. To illustrate the difficulties, we look at the classical "9-th roots of unity" and the distinction between the constructible zeroes and the others --- with respect to ruler and compass. However in all cases "infinite processes" are required that take us properly outside the realm of algebra.