# Tables of Number Fields with Prescribed Ramification

This site contains tables of number fields of low degree which are ramified at only a few small primes. Except where an entry is listed as "preliminary", we believe the tables to be exhaustive. Check back since the number of tables will expand over time.

If we fix a degree, n, and a finite set of primes, S, there are only finitely many fields of degree n and ramifying at only the primes in S. By exhaustive computer search, we have obtained what we believe to be complete lists for various combinations of n and S. We support two conventions with regard to the set S:

• list all fields which are unramified at all finite primes outside of S,
• list all fields which are unramified at all finite primes outside of S and which do ramify at every prime inside S. We will refer to this state as ramifying above S.

For example, we would say that the field defined by x² - 2 is ramified above S={2}, but that it is also unramified outside of S={2,3,7,41}.

In either case, we do not separate fields based on their ramification at infinity. Both conventions have their advantages, so you may view fields grouped either way.

Independent of this redundancy, we provide each table of fields is in two formats: dvi and gp/pari. File format information has been squirreled away to its own web page.

### Next, we have fields which ramify above S.

See also imprimitive number fields computed by Eric Driver.

## Links

Sites related to number theory are too numerous to list here. Try the Number Theory Web for a good collection.

One site of particular interest has content which is closely related to the content here: Bordeaux ftp site contains tables of number fields of low degree. The emphasis there are on fields with small absolute discriminant.

## Extras

Here are two related java programs I wrote.

• One piece of data included in the tables is, for each prime p in S, the exponent of p in the discriminant of the given field is listed. The maximum such exponent among all fields of a given degree is easy to compute. For example, one could use the Maximum Discriminant Exponent Calculator.
• How hard is it to produce tables such as the ones found above? The Targeted Hunter Search Calculator will estimate the minimum number of polynomials one would have to examine for a given table.

John Jones
Last modified: Fri Jun 29 20:00:02 MST 2001