# How are problems about number fields reduced to problems about their absolute Galois groups?

The article on Wikipedia about Neukirch–Uchida theorem claims right from the beginning the statement in my question. I have seen similar claims elsewhere before.

I am a little puzzled by this assertion. What they proved are the following(taken from the same article): 1."Jürgen Neukirch (1969) showed that two algebraic number fields with the same absolute Galois group are isomorphic". 2."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".

It is not clear to me that these prove that the entire algebraic number theory can be embedded into group theory. For instance, how can a question about the number of rational solutions of a Diophantine equation be answered by looking at just the absolute Galois group?

Obviously, explicit determination of Galois extensions of a number field is an important question in algebraic number theory, but that is not the only problem in the field.

Am I misinterpreting the assertions? If not, what do they mean by "all problems"? Is there a categorical statement?

• This blogpost discusses some how the study of the absolute Galois group can be applied to explicit "classical" problems (including the study of Diophantine equations).
– Mark
Sep 28, 2020 at 3:56
• You probably need an explicit version of that theorem. See kurims.kyoto-u.ac.jp/~yuichiro/rims1819revised.pdf Sep 28, 2020 at 6:25
• @Mark That is a very nice blog post. I have seen attractive phrases like "solving Diophantine equations by Galois representation" before. I can appreciate this perspective, and the explanation is convincing enough for practical purposes. However, powerful as they are, I don't think there is a metamathematical principle which guarantees that Galois representation is the silverbullet. My question is really asking for a reduction in the sense of logic. My impression is that such thing does not yet exist, or cannot exist. Sep 28, 2020 at 11:42