# Introduction

This is a web interface to the database of number fields of degree up to 15 created by Jürgen Klüners and Gunter Malle.

You must specify the degree of the extension. All other input fields are optional.

# Format of the output

The last column contains generating poynomials for all fields matching the search parameters. The second last column contains the field discriminant and the third last column its prime factorization.

# Description of the input format

The field maximal absolute value of the discriminant allows you to give an upper bound for the field discriminant. This may be interesting when otherwise the output gets too large.

With the field divisors of the discriminant you can specify divisors that should occur in the discriminant. Different divisors must be separated by commas, e.g.: 16, 3 but not 16; 3 or 16 3. The script will ignore input that does not match the required form.

The field all prime divisors of the discriminant allows to specify a list of primes. The factorization of the discriminant must not contain any other prime. For example if you enter 3 here, you will only see fields with a discriminant that is a power of 3.

If you are interested in fields such that the prime factorization of the discriminant contains only primes less than a fixed number you can use the next input form. Entering 20 here is equivalent to entering 2, 3, 5, 7, 11, 13, 17, 19 in the previous input form.

The last imput form allows you to specify the maximum number of different prime factors.

# Examples

If you enter:

 Degree of the polynomial 4 Type of the group (GAP, Magma) 3 Number of real zeros 0 Maximal absolute value of the discriminant 10^4 Divisors of the discriminant 1024 All prime divisors of the discriminant Maximal value for prime divisors of the discriminant 10 Number of prime divisors of the discriminant

You get :

 Degree 4 Group number 3 Signature 0 Maximal absolute value of the discriminant 10^4 Divisors of the discriminant 1024 All prime divisors are less than +10
 factorization discriminant generating polynomial 2^11 2048 x^4+2 2^10*3 3072 x^4+2*x^2+3 2^10*3 3072 x^4-2*x^2+3 2^11*3 6144 x^4+4*x^2+6 2^11*3 6144 x^4-4*x^2+6 2^10*7 7168 x^4+6*x^2+7
, the type of the Galois group (according to KASH or Magma) and the number of real zeros.