Neukirch–Uchida theorem

In mathematics, the Neukirch–Uchida theorem shows that all problems about algebraic number fields can be reduced to problems about their absolute Galois groups. Jürgen Neukirch (1969) showed that two algebraic number fields with the same absolute Galois group are isomorphic, and Kôji Uchida (1976) strengthened this by proving Neukirch's conjecture that automorphisms of the algebraic number field correspond to outer automorphisms of its absolute Galois group. Florian Pop (1990, 1994) 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

$\phi \colon \operatorname {Gal} ({\bar {K}}_{1}/K_{1})\xrightarrow {\cong } \operatorname {Gal} ({\bar {K}}_{2}/K_{2})$

of the absolute Galois groups, there exists a unique field isomorphism $\sigma \colon {\bar {K}}_{1}\xrightarrow {\cong } {\bar {K}}_{2}$ such that

$\sigma (K_{1})=K_{2}$

and

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

for every $g\in \operatorname {Gal} ({\bar {K}}_{1}/K_{1})$. The following diagram illustrates this condition.

${\begin{matrix}K_{1}&\hookrightarrow &{\bar {K_{1}}}&{\xrightarrow[{\cong }]{g}}&{\bar {K_{1}}}\\{\scriptstyle \sigma |_{K_{1}}}\downarrow &&\downarrow {\scriptstyle \sigma }&&\downarrow {\scriptstyle \sigma }\\K_{2}&\hookrightarrow &{\bar {K_{2}}}&{\xrightarrow[{\phi (g)}]{\cong }}&{\bar {K_{2}}}\end{matrix}}$

In particular, for algebraic number fields $K_{1}$, $K_{2}$, the following two conditions are equivalent.

• $\operatorname {Gal} ({\bar {K_{1}}}/K_{1})\cong \operatorname {Gal} ({\bar {K_{2}}}/K_{2})$
• $K_{1}\cong K_{2}$

References

• Neukirch, Jürgen (1969), "Kennzeichnung der p-adischen und der endlichen algebraischen Zahlkörper", Inventiones Mathematicae (in German), 6 (4): 296–314, Bibcode:1969InMat...6..296N, doi:10.1007/BF01425420, MR 0244211, S2CID 122002867
• Neukirch, Jürgen (1969), "Kennzeichnung der endlich-algebraischen Zahlkörper durch die Galoisgruppe der maximal auflösbaren Erweiterungen", Journal für die reine und angewandte Mathematik (in German), 238: 135–147, MR 0258804
• Uchida, Kôji (1976), "Isomorphisms of Galois groups.", J. Math. Soc. Jpn., 28 (4): 617–620, doi:10.2969/jmsj/02840617, MR 0432593
• Pop, Florian (1990), "On the Galois theory of function fields of one variable over number fields", Journal für die reine und angewandte Mathematik, 1990 (406): 200–218, doi:10.1515/crll.1990.406.200, MR 1048241, S2CID 119934490
• Pop, Florian (1994), "On Grothendieck's conjecture of birational anabelian geometry", Annals of Mathematics, (2), 139 (1): 145–182, doi:10.2307/2946630, JSTOR 2946630, MR 1259367