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Kummer theory
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<h2 id="kummer-extensions">Kummer extensions</h2> <p>A Kummer extension is a field extension <i>L</i>/<i>K</i>, where for some given integer <i>n</i> > 1 we have </p> <ul><li><i>K</i> contains <i>n</i> distinct <i>n</i>th <a href="/facts/Root_of_unity/EqH0ho1Y">roots of unity</a> (i.e., roots of <i>X</i><i>n</i> − 1)</li> <li><i>L</i>/<i>K</i> has <a href="/facts/Abelian_group/FfWwmx4R">abelian</a> <a href="/facts/Galois_group/A0p35p5c">Galois group</a> of <a href="/facts/Exponent_(group_theory)/TEvLhDtm">exponent</a> <i>n</i>.</li></ul> <p>For example, when <i>n</i> = 2, the first condition is always true if <i>K</i> has <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a> ≠ 2. The Kummer extensions in this case include quadratic extensions L = K ( a ) {\displaystyle L=K({\sqrt {a}})} where <i>a</i> in <i>K</i> is a non-square element. By the usual solution of <a href="/facts/Quadratic_equation/9IMnkJAT">quadratic equations</a>, any extension of degree 2 of <i>K</i> has this form. The Kummer extensions in this case also include biquadratic extensions and more general multiquadratic extensions. When <i>K</i> has characteristic 2, there are no such Kummer extensions. </p><p>Taking <i>n</i> = 3, there are no degree 3 Kummer extensions of the <a href="/facts/Rational_number/VJQmyY05">rational number</a> field Q, since for three cube roots of 1 <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> are required. If one takes <i>L</i> to be the splitting field of <i>X</i>3 − <i>a</i> over Q, where <i>a</i> is not a cube in the rational numbers, then <i>L</i> contains a subfield <i>K</i> with three cube roots of 1; that is because if α and β are roots of the cubic polynomial, we shall have (α/β)3 =1 and the cubic is a <a href="/facts/Separable_polynomial/5mcxhRGw">separable polynomial</a>. Then <i>L</i>/<i>K</i> is a Kummer extension. </p><p>More generally, it is true that when <i>K</i> contains <i>n</i> distinct <i>n</i>th roots of unity, which implies that the characteristic of <i>K</i> doesn't divide <i>n</i>, then adjoining to <i>K</i> the <i>n</i>th root of any element <i>a</i> of <i>K</i> creates a Kummer extension (of degree <i>m</i>, for some <i>m</i> dividing <i>n</i>). As the <a href="/facts/Splitting_field/m38eWUH1">splitting field</a> of the polynomial <i>Xn</i> − <i>a</i>, the Kummer extension is necessarily <a href="/facts/Galois_extension/52fzNQbv">Galois</a>, with Galois group that is <a href="/facts/Cyclic_group/JL5zfFuL">cyclic</a> of order <i>m</i>. It is easy to track the Galois action via the root of unity in front of a n . {\displaystyle {\sqrt[{n}]{a}}.} </p><p>Kummer theory provides converse statements. When <i>K</i> contains <i>n</i> distinct <i>n</i>th roots of unity, it states that any <a href="/facts/Abelian_extension/vIdW4Hnv">abelian extension</a> of <i>K</i> of exponent dividing <i>n</i> is formed by extraction of roots of elements of <i>K</i>. Further, if <i>K</i>× denotes the multiplicative group of non-zero elements of <i>K</i>, abelian extensions of <i>K</i> of exponent <i>n</i> correspond bijectively with subgroups of </p> K × / ( K × ) n , {\displaystyle K^{\times }/(K^{\times })^{n},} <p>that is, elements of <i>K</i>× <a href="/facts/Modular_arithmetic/0wtRa5dU">modulo</a> <i>n</i>th powers. The correspondence can be described explicitly as follows. Given a subgroup </p> Δ ⊆ K × / ( K × ) n , {\displaystyle \Delta \subseteq K^{\times }/(K^{\times })^{n},} <p>the corresponding extension is given by </p> K ( Δ 1 n ) , {\displaystyle K\left(\Delta ^{\frac {1}{n}}\right),} <p>where </p> Δ 1 n = { a n : a ∈ K × , a ⋅ ( K × ) n ∈ Δ } . {\displaystyle \Delta ^{\frac {1}{n}}=\left\{{\sqrt[{n}]{a}}:a\in K^{\times },a\cdot \left(K^{\times }\right)^{n}\in \Delta \right\}.} <p>In fact it suffices to adjoin <i>n</i>th root of one representative of each element of any set of generators of the group Δ. Conversely, if <i>L</i> is a Kummer extension of <i>K</i>, then Δ is recovered by the rule </p> Δ = ( K × ∩ ( L × ) n ) / ( K × ) n . {\displaystyle \Delta =\left(K^{\times }\cap (L^{\times })^{n}\right)/(K^{\times })^{n}.} <p>In this case there is an isomorphism </p> Δ ≅ Hom c ( Gal ( L / K ) , μ n ) {\displaystyle \Delta \cong \operatorname {Hom} _{\text{c}}(\operatorname {Gal} (L/K),\mu _{n})} <p>given by </p> a ↦ ( σ ↦ σ ( α ) α ) , {\displaystyle a\mapsto \left(\sigma \mapsto {\frac {\sigma (\alpha )}{\alpha }}\right),} <p>where α is any <i>n</i>th root of <i>a</i> in <i>L</i>. Here μ n {\displaystyle \mu _{n}} denotes the multiplicative group of <i>n</i>th roots of unity (which belong to <i>K</i>) and Hom c ( Gal ( L / K ) , μ n ) {\displaystyle \operatorname {Hom} _{\text{c}}(\operatorname {Gal} (L/K),\mu _{n})} is the group of continuous homomorphisms from Gal ( L / K ) {\displaystyle \operatorname {Gal} (L/K)} equipped with <a href="/facts/Krull_topology/0RX9Da9C">Krull topology</a> to μ n {\displaystyle \mu _{n}} with discrete topology (with group operation given by pointwise multiplication). This group (with discrete topology) can also be viewed as <a href="/facts/Pontryagin_dual/fbD3d8kz">Pontryagin dual</a> of Gal ( L / K ) {\displaystyle \operatorname {Gal} (L/K)} , assuming we regard μ n {\displaystyle \mu _{n}} as a subgroup of <a href="/facts/Circle_group/kuzdWtHo">circle group</a>. If the extension <i>L</i>/<i>K</i> is finite, then Gal ( L / K ) {\displaystyle \operatorname {Gal} (L/K)} is a finite discrete group and we have </p> Δ ≅ Hom ( Gal ( L / K ) , μ n ) ≅ Gal ( L / K ) , {\displaystyle \Delta \cong \operatorname {Hom} (\operatorname {Gal} (L/K),\mu _{n})\cong \operatorname {Gal} (L/K),} <p>however the last isomorphism isn't <a href="/facts/Natural_homomorphism/xwPumTHt">natural</a>. </p> <h3>Recovering <i>a</i>1/<i>n</i> from a primitive element</h3> <p>For p {\displaystyle p} prime, let K {\displaystyle K} be a field containing ζ p {\displaystyle \zeta _{p}} and K ( β ) / K {\displaystyle K(\beta )/K} a degree p {\displaystyle p} Galois extension. Note the Galois group is cyclic, generated by σ {\displaystyle \sigma } . Let </p> α = ∑ l = 0 p − 1 ζ p l σ l ( β ) ∈ K ( β ) {\displaystyle \alpha =\sum _{l=0}^{p-1}\zeta _{p}^{l}\sigma ^{l}(\beta )\in K(\beta )} <p>Then </p> ζ p σ ( α ) = ∑ l = 0 p − 1 ζ p l + 1 σ l + 1 ( β ) = α . {\displaystyle \zeta _{p}\sigma (\alpha )=\sum _{l=0}^{p-1}\zeta _{p}^{l+1}\sigma ^{l+1}(\beta )=\alpha .} <p>Since α ≠ σ ( α ) , K ( α ) = K ( β ) {\displaystyle \alpha \neq \sigma (\alpha ),K(\alpha )=K(\beta )} and </p> α p = ± ∏ l = 0 p − 1 ζ p − l α = ± ∏ l = 0 p − 1 σ l ( α ) = ± N K ( β ) / K ( α ) ∈ K {\displaystyle \alpha ^{p}=\pm \prod _{l=0}^{p-1}\zeta _{p}^{-l}\alpha =\pm \prod _{l=0}^{p-1}\sigma ^{l}(\alpha )=\pm N_{K(\beta )/K}(\alpha )\in K} , <p>where the ± {\displaystyle \pm } sign is + {\displaystyle +} if p {\displaystyle p} is odd and − {\displaystyle -} if p = 2 {\displaystyle p=2} . </p><p>When L / K {\displaystyle L/K} is an abelian extension of degree n = ∏ j = 1 m p j {\displaystyle n=\prod _{j=1}^{m}p_{j}} square-free such that ζ n ∈ K {\displaystyle \zeta _{n}\in K} , apply the same argument to the subfields K ( β j ) / K {\displaystyle K(\beta _{j})/K} Galois of degree p j {\displaystyle p_{j}} to obtain </p> L = K ( a 1 1 / p 1 , … , a m 1 / p m ) = K ( A 1 / p 1 , … , A 1 / p m ) = K ( A 1 / n ) {\displaystyle L=K\left(a_{1}^{1/p_{1}},\ldots ,a_{m}^{1/p_{m}}\right)=K\left(A^{1/p_{1}},\ldots ,A^{1/p_{m}}\right)=K\left(A^{1/n}\right)} <p>where </p> A = ∏ j = 1 m a j n / p j ∈ K {\displaystyle A=\prod _{j=1}^{m}a_{j}^{n/p_{j}}\in K} . <h2 id="the-kummer-map">The Kummer Map</h2> <p>One of the main tools in Kummer theory is the Kummer map. Let m {\displaystyle m} be a positive integer and let K {\displaystyle K} be a field, not necessarily containing the m {\displaystyle m} th roots of unity. Letting K ¯ {\displaystyle {\overline {K}}} denote the algebraic closure of K {\displaystyle K} , there is a <a href="/facts/Short_exact_sequence/leUJnwJb">short exact sequence</a> </p><p> 0 → K ¯ × [ m ] → K ¯ × → z ↦ z m K ¯ × → 0 {\displaystyle 0\xrightarrow {} {\overline {K}}^{\times }[m]\xrightarrow {} {\overline {K}}^{\times }\xrightarrow {z\mapsto z^{m}} {\overline {K}}^{\times }\xrightarrow {} 0} </p><p>Choosing an extension L / K {\displaystyle L/K} and taking G a l ( K ¯ / L ) {\displaystyle \mathrm {Gal} ({\overline {K}}/L)} -cohomology one obtains the sequence </p><p> 0 → L × / ( L × ) m → H 1 ( L , K ¯ × [ m ] ) → H 1 ( L , K ¯ × ) [ m ] → 0 {\displaystyle 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} </p><p>By <a href="/facts/Hilbert%27s_Theorem_90/jKsTzlzN">Hilbert's Theorem 90</a> H 1 ( L , K ¯ × ) = 0 {\displaystyle H^{1}\left(L,{\overline {K}}^{\times }\right)=0} , and hence we get an isomorphism δ : L × / ( L × ) m → ∼ H 1 ( L , K ¯ × [ m ] ) {\displaystyle \delta :L^{\times }/\left(L^{\times }\right)^{m}\xrightarrow {\sim } H^{1}\left(L,{\overline {K}}^{\times }[m]\right)} . This is the Kummer map. A version of this map also exists when all m {\displaystyle m} are considered simultaneously. Namely, since L × / ( L × ) m = L × ⊗ m − 1 Z / Z {\displaystyle L^{\times }/(L^{\times })^{m}=L^{\times }\otimes m^{-1}\mathbb {Z} /\mathbb {Z} } , taking the direct limit over m {\displaystyle m} yields an isomorphism </p><p> δ : L × ⊗ Q / Z → ∼ H 1 ( L , K ¯ t o r s ) {\displaystyle \delta :L^{\times }\otimes \mathbb {Q} /\mathbb {Z} \xrightarrow {\sim } H^{1}\left(L,{\overline {K}}_{tors}\right)} , </p><p>where <i>tors</i> denotes the torsion subgroup of roots of unity. </p> <h2 id="for-elliptic-curves">For Elliptic Curves</h2> <p>Kummer theory is often used in the context of elliptic curves. Let E / K {\displaystyle E/K} be an elliptic curve. There is a short exact sequence </p><p> 0 → E [ m ] → E → P ↦ m ⋅ P E → 0 {\displaystyle 0\xrightarrow {} E[m]\xrightarrow {} E\xrightarrow {P\mapsto m\cdot P} E\xrightarrow {} 0} , </p><p>where the multiplication by m {\displaystyle m} map is surjective since E {\displaystyle E} is divisible. Choosing an algebraic extension L / K {\displaystyle L/K} and taking cohomology, we obtain the Kummer sequence for E {\displaystyle E} : </p><p> 0 → E ( L ) / m E ( L ) → H 1 ( L , E [ m ] ) → H 1 ( L , E ) [ m ] → 0 {\displaystyle 0\xrightarrow {} E(L)/mE(L)\xrightarrow {} H^{1}(L,E[m])\xrightarrow {} H^{1}(L,E)[m]\xrightarrow {} 0} . </p><p>The computation of the weak Mordell-Weil group E ( L ) / m E ( L ) {\displaystyle E(L)/mE(L)} is a key part of the proof of the <a href="/facts/Mordell%E2%80%93Weil_theorem/3ko0yKew">Mordell-Weil theorem</a>. The failure of H 1 ( L , E ) {\displaystyle H^{1}(L,E)} to vanish adds a key complexity to the theory. </p> <h2 id="generalizations">Generalizations</h2> <p>Suppose that <i>G</i> is a <a href="/facts/Profinite_group/0RX9Da9C">profinite group</a> acting on a module <i>A</i> with a surjective homomorphism π from the <i>G</i>-module <i>A</i> to itself. Suppose also that <i>G</i> acts trivially on the kernel <i>C</i> of π and that the first cohomology group H1(<i>G</i>,<i>A</i>) is trivial. Then the exact sequence of group cohomology shows that there is an isomorphism between <i>A</i><i>G</i>/π(<i>A</i><i>G</i>) and Hom(<i>G</i>,<i>C</i>). </p><p>Kummer theory is the special case of this when <i>A</i> is the multiplicative group of the separable closure of a field <i>k</i>, <i>G</i> is the Galois group, π is the <i>n</i>th power map, and <i>C</i> the group of <i>n</i>th roots of unity. <a href="/facts/Artin%E2%80%93Schreier_theory/nC8hvw8k">Artin–Schreier theory</a> is the special case when <i>A</i> is the additive group of the separable closure of a field <i>k</i> of positive characteristic <i>p</i>, <i>G</i> is the Galois group, π is the <a href="/facts/Frobenius_map/b9hkmc97">Frobenius map</a> minus the identity, and <i>C</i> the finite field of order <i>p</i>. Taking <i>A</i> to be a ring of truncated Witt vectors gives Witt's generalization of Artin–Schreier theory to extensions of exponent dividing <i>pn</i>. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Quadratic_field/dUnSDN4e">Quadratic field</a></li></ul> <ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=Kummer_extension">"Kummer extension"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li> <li><a href="/facts/Bryan_John_Birch/QVWLhz0w">Bryan Birch</a>, "Cyclotomic fields and Kummer extensions", in <a href="/facts/J.W.S._Cassels/AC33eVoA">J.W.S. Cassels</a> and <a href="/facts/A._Frohlich/dSb1kUcs">A. Frohlich</a> (edd), <i>Algebraic number theory</i>, <a href="/facts/Academic_Press/qxoKjzFy">Academic Press</a>, 1973. Chap.III, pp. 85–93.</li></ul>