File Formats

Dvi files

Dvi files are output by TeX, or in this case LaTeX. If you are interested in these tables but find it difficult to preview them, e-mail the maintainer and we can see about converting the tables to another format.

Now for the format of the tables themselves. The fields are first sorted by discriminant class the square-free part of the field discriminant. Among fields with the same root field, we sort by the order of the Galois group (of the Galois closure), and then by absolute discriminant (i.e., absolute value of the field discriminant). Some tables have extra sorting, which is explained on the corresponding web pages containing links to the fields.

Each field is allocated (at least) two lines in its table, and vertical lines separate the entries into blocks. Hanging off of the left side of each table are line numbers. These are intended to make it easy to refer to particular fields from the tables.

• The first block contains global information:
  • the top line shows Artin exponents for each prime in S and for the infinite place. Each exponent is subscripted with the root number associated to that place. The root numbers are 4th roots of unity; we write "+" for 1, "-" for -1, "i" for i, and "k" for -i. Root numbers for unramified primes are equal to 1. The product of all of the root numbers for a given field must be 1.
  • the bottom line lists the Galois group of the normal closure of the field, followed by the class number (h). If the class number is 1, we leave that entry blank.
• The next block lists the coefficients of a defining polynomial: xⁿ+ a₁x^n-1 + . . . + a_n.
• Next, there is a series of blocks with local information for each prime p in S. If we tensor our field with Q_p, the local algebra is isomorphic to a product of fields, each a finite extension of Q_p. We devote one line to each such field not equal to Q_p. We order these rows first by degree, then by ramification degree, and break any remaining ties by Artin exponent.

  For each non-trivial p-adic completion, we display:

  • f^e where f is the residual degree and e is the ramification index. If e=1, we omit the exponent.
  • In the column labeled c_w we list the Artin exponent subscripted by the root number for the local field.
  • The entry marked d gives the discriminant class of the local field. Rather than picking a square-free p-adic integer, we adopt the following convention: if p is an odd prime, we use either 1, p, *, or p* where "*" represents any non-square unit in Z_p. Similarly, we represent the eight separable quadratic algebras over Q₂ by 1, -1, 2, -2, and their "* forms".
  • The column marked s displays information about quadratic subfields. If there is a single quadratic subfield, it is displayed with the same convention as root fields. If there are 3 such subfields, we print V when the p is odd since their compositum is the unique V₄ extension of Q_p. When p = 2, there are 7 V₄ extensions, so instead we print the codes for two of the quadratic subfields (one stacked above the other).

  Note, some of the data listed in the first block can be deduced from these local entries. In particular, Artin exponents add and root numbers multiply to give the corresponding entry in first block. Moreover, discriminant classes are multiplicative.

• The remaining large block shows splitting for small primes not in S as a product of residual degrees. We omit entries of 1. Hence, if a prime splits completely, the entry is blank.

Pari files

Pari is a freely available package for doing mathematical computations, especially number theoretic ones. Gp is a command line "calculator" based on the Pari programming library. We use the terms here interchangeably.

At the moment, gp files of number fields will simply be a list of all polynomials. The philosophy is that it is not worth trying to guess exactly which invariants someone may want considering how easy it is to have gp calculate them. The entries in the gp files are in the same order as the corresponding dvi file. Thus, if you see that field number 12 looks interesting from one of the dvi files, you should be able to access it as the 12th entry in the gp file.

Note, these files are plain text files. All operations are explicitly listed, so one should be able to import them directly into Maple as well. To use with Mathematica, one would have to change two symbols: the opening and closing square brackets which delimit the ends of the lists of the polynomials should be converted into curly braces to make it a Mathematica list.