Small primes of degree one#
Iterator for finding several primes of absolute degree one of a number field of small prime norm.
ALGORITHM:
Let \(P\) denote the product of some set of prime numbers. (In practice, we use the product of the first 10000 primes, because Pari computes this many by default.)
Let \(K\) be a number field and let \(f(x)\) be a polynomial defining \(K\) over the rational field. Let \(\alpha\) be a root of \(f\) in \(K\).
We know that \([ O_K : \ZZ[\alpha] ]^2 = | \Delta(f(x)) / \Delta(O_K) |\), where \(\Delta\) denotes the discriminant (see, for example, Proposition 4.4.4, p165 of [Coh1993]). Therefore, after discarding primes dividing \(\Delta(f(x))\) (this includes all ramified primes), any integer \(n\) such that \(\gcd(f(n), P) > 0\) yields a prime \(p | P\) such that \(f(x)\) has a root modulo \(p\). By the condition on discriminants, this root is a single root. As is well known (see, for example Theorem 4.8.13, p199 of [Coh1993]), the ideal generated by \((p, \alpha - n)\) is prime and of degree one.
Warning
It is possible that there are no primes of \(K\) of absolute degree one of small prime norm, and it is possible that this algorithm will not find any primes of small norm.
Todo
There are situations when this will fail. There are questions of finding primes of relative degree one. There are questions of finding primes of exact degree larger than one. In short, if you can contribute, please do!
EXAMPLES:
sage: x = ZZ['x'].gen()
sage: F.<a> = NumberField(x^2 - 2)
sage: Ps = F.primes_of_degree_one_list(3)
sage: Ps # random
[Fractional ideal (2*a + 1), Fractional ideal (-3*a + 1), Fractional ideal (-a + 5)]
sage: [ P.norm() for P in Ps ] # random
[7, 17, 23]
sage: all(ZZ(P.norm()).is_prime() for P in Ps)
True
sage: all(P.residue_class_degree() == 1 for P in Ps)
True
>>> from sage.all import *
>>> x = ZZ['x'].gen()
>>> F = NumberField(x**Integer(2) - Integer(2), names=('a',)); (a,) = F._first_ngens(1)
>>> Ps = F.primes_of_degree_one_list(Integer(3))
>>> Ps # random
[Fractional ideal (2*a + 1), Fractional ideal (-3*a + 1), Fractional ideal (-a + 5)]
>>> [ P.norm() for P in Ps ] # random
[7, 17, 23]
>>> all(ZZ(P.norm()).is_prime() for P in Ps)
True
>>> all(P.residue_class_degree() == Integer(1) for P in Ps)
True
The next two examples are for relative number fields.:
sage: L.<b> = F.extension(x^3 - a)
sage: Ps = L.primes_of_degree_one_list(3)
sage: Ps # random
[Fractional ideal (17, b - 5), Fractional ideal (23, b - 4), Fractional ideal (31, b - 2)]
sage: [ P.absolute_norm() for P in Ps ] # random
[17, 23, 31]
sage: all(ZZ(P.absolute_norm()).is_prime() for P in Ps)
True
sage: all(P.residue_class_degree() == 1 for P in Ps)
True
sage: M.<c> = NumberField(x^2 - x*b^2 + b)
sage: Ps = M.primes_of_degree_one_list(3)
sage: Ps # random
[Fractional ideal (17, c - 2), Fractional ideal (c - 1), Fractional ideal (41, c + 15)]
sage: [ P.absolute_norm() for P in Ps ] # random
[17, 31, 41]
sage: all(ZZ(P.absolute_norm()).is_prime() for P in Ps)
True
sage: all(P.residue_class_degree() == 1 for P in Ps)
True
>>> from sage.all import *
>>> L = F.extension(x**Integer(3) - a, names=('b',)); (b,) = L._first_ngens(1)
>>> Ps = L.primes_of_degree_one_list(Integer(3))
>>> Ps # random
[Fractional ideal (17, b - 5), Fractional ideal (23, b - 4), Fractional ideal (31, b - 2)]
>>> [ P.absolute_norm() for P in Ps ] # random
[17, 23, 31]
>>> all(ZZ(P.absolute_norm()).is_prime() for P in Ps)
True
>>> all(P.residue_class_degree() == Integer(1) for P in Ps)
True
>>> M = NumberField(x**Integer(2) - x*b**Integer(2) + b, names=('c',)); (c,) = M._first_ngens(1)
>>> Ps = M.primes_of_degree_one_list(Integer(3))
>>> Ps # random
[Fractional ideal (17, c - 2), Fractional ideal (c - 1), Fractional ideal (41, c + 15)]
>>> [ P.absolute_norm() for P in Ps ] # random
[17, 31, 41]
>>> all(ZZ(P.absolute_norm()).is_prime() for P in Ps)
True
>>> all(P.residue_class_degree() == Integer(1) for P in Ps)
True
AUTHORS:
Nick Alexander (2008): initial version
David Loeffler (2009): fixed a bug with relative fields
Maarten Derickx (2017): fixed a bug with number fields not generated by an integral element
- class sage.rings.number_field.small_primes_of_degree_one.Small_primes_of_degree_one_iter(field, num_integer_primes=10000, max_iterations=100)[source]#
Bases:
object
Iterator that finds primes of a number field of absolute degree one and bounded small prime norm.
INPUT:
field
– aNumberField
.num_integer_primes
– (default: 10000) an integer. We try to find primes of absolute norm no greater than thenum_integer_primes
-th prime number. For example, ifnum_integer_primes
is 2, the largest norm found will be 3, since the second prime is 3.max_iterations
– (default: 100) an integer. We testmax_iterations
integers to find small primes before raisingStopIteration
.
AUTHOR:
Nick Alexander
- next()[source]#
Return a prime of absolute degree one of small prime norm.
Raises
StopIteration
if such a prime cannot be easily found.EXAMPLES:
sage: x = QQ['x'].gen() sage: K.<a> = NumberField(x^2 - 3) sage: it = K.primes_of_degree_one_iter() sage: [ next(it) for i in range(3) ] # random [Fractional ideal (2*a + 1), Fractional ideal (-a + 4), Fractional ideal (3*a + 2)]
>>> from sage.all import * >>> x = QQ['x'].gen() >>> K = NumberField(x**Integer(2) - Integer(3), names=('a',)); (a,) = K._first_ngens(1) >>> it = K.primes_of_degree_one_iter() >>> [ next(it) for i in range(Integer(3)) ] # random [Fractional ideal (2*a + 1), Fractional ideal (-a + 4), Fractional ideal (3*a + 2)]