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Algebraic number fields are determined by their absolute Galois groups
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.
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