 Finite-extension kernel of a Kummer map

ID: finite-extension-kernel-of-a-kummer-map

L / K

, the kernel of

K^{*} / (K^{*})^{n} ⟶ L^{*} / (L^{*})^{n}

injects into the finite group

H^{1} (Gal (L / K), μ_{n} (L))

. Likewise, the kernel of

E (K) / n E (K) \to E (L) / n E (L)

injects into

H^{1} (Gal (L / K), E [n] (L))

and is finite.

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