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Kummer theory
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== The Kummer Map == One of the main tools in Kummer theory is the Kummer map. Let <math>m</math> be a positive integer and let <math>K</math> be a field, not necessarily containing the <math>m</math>th roots of unity. Letting <math>\overline{K}</math> denote the algebraic closure of <math>K</math>, there is a [[short exact sequence]] <math>0\xrightarrow{} \overline{K}^{\times}[m] \xrightarrow{} \overline{K}^{\times} \xrightarrow{z\mapsto z^m} \overline{K}^{\times}\xrightarrow{} 0</math> Choosing an extension <math>L/K</math> and taking <math>\mathrm{Gal}(\overline{K}/L)</math>-cohomology one obtains the sequence <math>0\xrightarrow{} L^{\times}/(L^{\times})^{m} \xrightarrow{} H^1\left(L, \overline{K}^{\times}[m]\right) \xrightarrow{} H^1\left(L,\overline{K}^{\times}\right)[m]\xrightarrow{}0</math> By [[Hilbert's Theorem 90]] <math>H^1\left(L,\overline{K}^{\times}\right)=0</math>, and hence we get an isomorphism <math>\delta: L^{\times}/\left(L^{\times}\right)^m\xrightarrow{\sim}H^1\left(L,\overline{K}^{\times}[m]\right)</math>. This is the Kummer map. A version of this map also exists when all <math>m</math> are considered simultaneously. Namely, since <math>L^{\times}/(L^{\times})^m=L^{\times}\otimes m^{-1} \mathbb{Z}/\mathbb{Z}</math>, taking the direct limit over <math>m</math> yields an isomorphism <math>\delta: L^{\times} \otimes \mathbb{Q}/\mathbb{Z} \xrightarrow{\sim} H^1\left(L, \overline{K}_{tors}\right) </math>, where ''tors'' denotes the torsion subgroup of roots of unity.
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