Chapter 39: Cyclotomic Fields and Roots of Unity

The CRT isomorphism ℤ_τ ≅ ∏_p ℤ_p

decomposes the boundary ring into local factors, and the spectral characters χ_+, χ_- (see Part X, the relevant definition, I.D37) probe each factor. This chapter identifies where those characters live: the cyclotomic fields ℚ^cyc_τ(ζ_n), extensions of ℚ_τ generated by roots of unity. The Galois groups Gal(ℚ_τ(ζ_n)/ℚ_τ) ≅ (ℤ/nℤ)^× connect the CRT machinery to field automorphisms, previewing the Galois theory of Book II.