mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, 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 f ...

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'' tha ...

s. showed that two algebraic number fields with the same absolute Galois group are

isomorphic In mathematics, an isomorphism is a structure-preserving mapping 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 them. The word is ...

, 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 t ...

s of its absolute Galois group. extended the result to infinite fields that are finitely generated over

prime field In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive iden ...

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 restore ''X''. The first results for num ...

, whose main theme is to reduce properties of geometric objects to properties of their

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

s, provided these fundamental groups are sufficiently non-abelian.

References

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