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 nonprimitive
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 \(f(x) = x^n + a_{n1} x^{n1} + \dots + a_0\). Hunter’s theorem gives bounds on \(a_{n1}\) and \(a_{n2}\); then given \(a_{n1}\) and \(a_{n2}\), one can recursively compute bounds on \(a_{n3}, \dots, a_0\), 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!).
See also [Mar1980].
Authors¶
 John Voight (20070901): Initial version.
 John Voight (20070919): Various optimization tweaks.
 John Voight (20071009): Added DSage module.
 John Voight (20071017): Added pari functions to avoid recomputations.
 John Voight (20071027): Separated DSage component.
 Craig Citro and John Voight (20071104): Additional doctests and type checking.
 Craig Citro and John Voight (20080210): 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^(nd)
where
length(a) = d+1
, so in particular alwaysa[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 degreeB
– (integer) the discriminant bounda
– (list, default: []) the coefficient list to begin withverbose
– (integer or string, default: 0) ifverbose == 1
(or2
), 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) IfTrue
, 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) Ifkeep_fields
is True, then keep fields up toB*log(B)
; ifkeep_fields
is an integer, then keep fields up to that integer.t_2
– (boolean or integer, default: False) Ift_2 = T
, then keep only polynomials with t_2 norm >= T.just_print
– (boolean, default: False): ifjust_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 byjust_print
. In this case, we don’t return anything.return_pari_objects
– (boolean, default: True) if bothreturn_seqs
andreturn_pari_objects
areFalse
then it returns the elements as Sage objects; otherwise it returns pari objects.
OUTPUT:
the list of fields with entries
[d,f]
, whered
is the discriminant andf
is a defining polynomial, sorted by discriminant.AUTHORS:
 John Voight (20070903)
 Craig Citro (20080919): 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 (20070903)
NOTES: 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^23*x+1')],[5,pari('x^25')]] sage: sage.rings.number_field.totallyreal.weed_fields(ls) 1 sage: ls [[5, x^2  3*x + 1]]