Neukirch–Uchida Theorem
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Neukirch–Uchida theorem shows that all problems about
algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
s can be reduced to problems about their
absolute Galois group In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' ...
s. showed that two algebraic number fields with the same absolute Galois group are
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
, and strengthened this by proving Neukirch's conjecture that automorphisms of the algebraic number field correspond to
outer automorphism In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a ...
s of its absolute Galois group. extended the result to infinite fields that are finitely generated over
prime field In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers. A field is thus a fundamental algebraic structure which is wid ...
s. The Neukirch–Uchida theorem is one of the foundational results of
anabelian geometry Anabelian geometry is a theory in number theory which describes the way in which the algebraic fundamental group ''G'' of a certain arithmetic variety ''X'', or some related geometric object, can help to recover ''X''. The first results for nu ...
, whose main theme is to reduce properties of geometric objects to properties of their
fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
s, provided these fundamental groups are sufficiently non-abelian.


Statement

Let K_1, K_2 be two
algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
s. The Neukirch–Uchida theorem says that, for every topological group
isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
:\phi\colon\operatorname(\bar K_1/K_1)\xrightarrow\cong\operatorname(\bar K_2/K_2) of the
absolute Galois group In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' ...
s, 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^ for every g\in\operatorname(\bar K_1/K_1). The following diagram illustrates this condition. :\begin K_1 & \hookrightarrow & \bar & \xrightarrow cong & \bar \\ \downarrow & & \downarrow & & \downarrow \\ K_2 & \hookrightarrow & \bar & \xrightarrow
phi(g) Phi ( ; uppercase Φ, lowercase φ or ϕ; ''pheî'' ; Modern Greek: ''fi'' ) is the twenty-first letter of the Greek alphabet. In Archaic and Classical Greek (c. 9th to 4th century BC), it represented an aspirated voiceless bilabial plos ...
& \bar \end In particular, for
algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
s K_1, K_2, the following two conditions are equivalent. * \operatorname(\bar/K_1)\cong\operatorname(\bar/K_2) * K_1\cong K_2


References

* * * * * Theorems in algebraic number theory {{abstract-algebra-stub