Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I know that finite simple groups have been exhaustively classified. Have fields (in the sense of corpora, not of vector fields) been exhaustively classified?

share|cite|improve this question
For example, algebraic number fields have not been exhaustively classified. – Makoto Kato Dec 31 '12 at 21:23
The obvious analogue, finite fields, have been classified. – arsmath Dec 31 '12 at 22:23
up vote 5 down vote accepted

To expand on Makoto Kato's comment. A finite field extension of the rationals is called a number field. There are various objects associated to number fields, such as their ring of integers and ideal class groups. The ideal class group in some sense measures how badly unique factorization fails in a number field's ring of integers. One open question in algebraic number theory is whether or not there exist infinitely many number fields with trivial class group. For Galois number fields we can also talk about their Galois group. Another open question is whether or not for every finite group $G$ we can find a number field such that its Galois group is $G$. So in this sense even number fields have not been exhaustively classified.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.