Neukirch–Uchida theorem

Template:Short description In mathematics, the Neukirch–Uchida theorem shows that all problems about algebraic number fields can be reduced to problems about their absolute Galois groups. Template:Harvs showed that two algebraic number fields with the same absolute Galois group are isomorphic, and Template:Harvs strengthened this by proving Neukirch's conjecture that automorphisms of the algebraic number field correspond to outer automorphisms of its absolute Galois group. Template:Harvs extended the result to infinite fields that are finitely generated over prime fields.

The Neukirch–Uchida theorem is one of the foundational results of anabelian geometry, whose main theme is to reduce properties of geometric objects to properties of their fundamental groups, provided these fundamental groups are sufficiently non-abelian.

Statement

Let $K_1$, $K_2$ be two algebraic number fields. The Neukirch–Uchida theorem says that, for every topological group isomorphism

$\varphi :\mathrm{Gal}({\overline{K}}_1/K_1)\stackrel{\cong }{\to }\mathrm{Gal}({\overline{K}}_2/K_2)$

of the absolute Galois groups, there exists a unique field isomorphism $\sigma :{\overline{K}}_1\stackrel{\cong }{\to }{\overline{K}}_2$ such that

$\sigma (K_1)=K_2$

and

$\varphi (g)=\sigma \circ g\circ {\sigma }^{-1}$

for every $g\in \mathrm{Gal}({\overline{K}}_1/K_1)$. The following diagram illustrates this condition.

$\begin{array}{ccccc}{K}_1& \hookrightarrow & \overline{K_1}& \underset{\cong }{\overset{g}{\to }}& \overline{K_1}\\ \sigma {|}_{K_1}\downarrow & & \downarrow \sigma & & \downarrow \sigma \\ K_2& \hookrightarrow & \overline{K_2}& \underset{\varphi (g)}{\overset{\cong }{\to }}& \overline{K_2}\end{array}$

In particular, for algebraic number fields $K_1$, $K_2$, the following two conditions are equivalent.

• $\mathrm{Gal}(\overline{K_1}/K_1)\cong \mathrm{Gal}(\overline{K_2}/K_2)$
• $K_1\cong K_2$

References

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