# Enumeration of Primitive Totally Real Fields#

This module contains functions for enumerating all primitive totally real number fields of given degree and small discriminant. Here a number field is called primitive if it contains no proper subfields except $$\QQ$$.

See also sage.rings.number_field.totallyreal_rel, which handles the non-primitive case using relative extensions.

## Algorithm#

We use Hunter’s algorithm ([Coh2000], Section 9.3) with modifications due to Takeuchi [Tak1999] and the author [Voi2008].

We enumerate polynomials $$f(x) = x^n + a_{n-1} x^{n-1} + \dots + a_0$$. Hunter’s theorem gives bounds on $$a_{n-1}$$ and $$a_{n-2}$$; then given $$a_{n-1}$$ and $$a_{n-2}$$, one can recursively compute bounds on $$a_{n-3}, \dots, a_0$$, using the fact that the polynomial is totally real by looking at the zeros of successive derivatives and applying Rolle’s theorem. See [Tak1999] for more details.

## Examples#

In this first simple example, we compute the totally real quadratic fields of discriminant $$\le 50$$.

sage: enumerate_totallyreal_fields_prim(2,50)
[[5, x^2 - x - 1],
[8, x^2 - 2],
[12, x^2 - 3],
[13, x^2 - x - 3],
[17, x^2 - x - 4],
[21, x^2 - x - 5],
[24, x^2 - 6],
[28, x^2 - 7],
[29, x^2 - x - 7],
[33, x^2 - x - 8],
[37, x^2 - x - 9],
[40, x^2 - 10],
[41, x^2 - x - 10],
[44, x^2 - 11]]
sage: [ d for d in range(5,50) if (is_squarefree(d) and d%4 == 1) or (d%4 == 0 and is_squarefree(d/4)) ]
[5, 8, 12, 13, 17, 20, 21, 24, 28, 29, 33, 37, 40, 41, 44]


Next, we compute all totally real quintic fields of discriminant $$\le 10^5$$:

sage: ls = enumerate_totallyreal_fields_prim(5,10^5) ; ls
[[14641, x^5 - x^4 - 4*x^3 + 3*x^2 + 3*x - 1],
[24217, x^5 - 5*x^3 - x^2 + 3*x + 1],
[36497, x^5 - 2*x^4 - 3*x^3 + 5*x^2 + x - 1],
[38569, x^5 - 5*x^3 + 4*x - 1],
[65657, x^5 - x^4 - 5*x^3 + 2*x^2 + 5*x + 1],
[70601, x^5 - x^4 - 5*x^3 + 2*x^2 + 3*x - 1],
[81509, x^5 - x^4 - 5*x^3 + 3*x^2 + 5*x - 2],
[81589, x^5 - 6*x^3 + 8*x - 1],
[89417, x^5 - 6*x^3 - x^2 + 8*x + 3]]
sage: len(ls)
9


We see that there are 9 such fields (up to isomorphism!).

## Authors#

• John Voight (2007-09-01): Initial version.

• John Voight (2007-09-19): Various optimization tweaks.

• John Voight (2007-10-09): Added DSage module.

• John Voight (2007-10-17): Added pari functions to avoid recomputations.

• John Voight (2007-10-27): Separated DSage component.

• Craig Citro and John Voight (2007-11-04): Additional doctests and type checking.

• Craig Citro and John Voight (2008-02-10): Final modifications for submission.

sage.rings.number_field.totallyreal.enumerate_totallyreal_fields_prim(n, B, a=[], verbose=0, return_seqs=False, phc=False, keep_fields=False, t_2=False, just_print=False, return_pari_objects=True)#

This function enumerates primitive totally real fields of degree $$n>1$$ with discriminant $$d \leq B$$; optionally one can specify the first few coefficients, where the sequence $$a$$ corresponds to

a[d]*x^n + ... + a[0]*x^(n-d)


where length(a) = d+1, so in particular always a[d] = 1.

Note

This is guaranteed to give all primitive such fields, and seems in practice to give many imprimitive ones.

INPUT:

• n – (integer) the degree

• B – (integer) the discriminant bound

• a – (list, default: []) the coefficient list to begin with

• verbose – (integer or string, default: 0) if verbose == 1 (or 2), then print to the screen (really) verbosely; if verbose is a string, then print verbosely to the file specified by verbose.

• return_seqs – (boolean, default False) If True, then return the polynomials as sequences (for easier exporting to a file).

• phc – boolean or integer (default: False)

• keep_fields – (boolean or integer, default: False) If keep_fields is True, then keep fields up to B*log(B); if keep_fields is an integer, then keep fields up to that integer.

• t_2 – (boolean or integer, default: False) If t_2 = T, then keep only polynomials with t_2 norm >= T.

• just_print – (boolean, default: False): if just_print is not False, instead of creating a sorted list of totally real number fields, we simply write each totally real field we find to the file whose filename is given by just_print. In this case, we don’t return anything.

• return_pari_objects – (boolean, default: True) if both return_seqs and return_pari_objects are False then it returns the elements as Sage objects; otherwise it returns pari objects.

OUTPUT:

the list of fields with entries [d,f], where d is the discriminant and f is a defining polynomial, sorted by discriminant.

AUTHORS:

• John Voight (2007-09-03)

• Craig Citro (2008-09-19): moved to Cython for speed improvement

sage.rings.number_field.totallyreal.odlyzko_bound_totallyreal(n)#

This function returns the unconditional Odlyzko bound for the root discriminant of a totally real number field of degree n.

Note

The bounds for n > 50 are not necessarily optimal.

INPUT:

• n (integer) the degree

OUTPUT:

a lower bound on the root discriminant (as a real number)

EXAMPLES:

sage: [sage.rings.number_field.totallyreal.odlyzko_bound_totallyreal(n) for n in range(1,5)]
[1.0, 2.223, 3.61, 5.067]


AUTHORS:

• John Voight (2007-09-03)

Note

The values are calculated by Martinet [Mar1980].

sage.rings.number_field.totallyreal.weed_fields(S, lenS=0)#

Function used internally by the enumerate_totallyreal_fields_prim() routine. (Weeds the fields listed by [discriminant, polynomial] for isomorphism classes.) Returns the size of the resulting list.

EXAMPLES:

sage: ls = [[5,pari('x^2-3*x+1')],[5,pari('x^2-5')]]
sage: sage.rings.number_field.totallyreal.weed_fields(ls)
1
sage: ls
[[5, x^2 - 3*x + 1]]