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?