Math 254a: Fermat Revisited
نویسنده
چکیده
Proof. Define a map U → μn by u 7→ u/ū; this gives a well-defined map by exercise 2 of homework 5. Compose with the quotient μn → μn/μ 2 n, and denote the resulting map by φ : U → μn/μ 2 n. It is enough to show that ker φ = μnU , since μn/μ 2 n has order 2. Certainly μnU + ⊂ kerφ, since φ(ζu) = [ζ] = 0 in μn/μ 2 n for u + ∈ U and ζ ∈ μn. Conversely, suppose φ(u) = 0 for some u ∈ U : then u/ū = ζ 2 for some ζ ∈ μn, and ζ u satisfies ζu/ζ̄ū = 1, so must be real, and u ∈ μnU , as desired.
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Math 254a: Density Theorems via Class Field Theory, and Fermat’s Last Theorem Revisited
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