(参考) https://www.uio.no/studier/emner/matnat/math/MAT4250/h13/ Universitetet i Oslo Semesterside for MAT4250 - Host 2013 Notes Cyclotomic fields https://www.uio.no/studier/emner/matnat/math/MAT4250/h13/cyclotomic.pdf Cyclotomic fields Preliminary version. Version 1+∞ - 22. oktober 2013 klokken P4 Proposition 2 If n and m are relatively prime natural numbers, then the two cyclotomic fields Q(ξn) and Q(ξm) are linearly disjoint. Their composite Q(ξn, ξm)is equal to Q(ξnm), and Q(ξn) ∩ Q(ξm) = Q. Proof: Clearly the composite of Q(ξn) and Q(ξm) contains Q(ξnm), the product ξnξm being a primitive nm-th root of unity. The Euler φ-function is multiplicative, so [Q(ξnm) : Q]=[Q(ξn) : Q][Q(ξm) : Q], and we are done.
