Routines for Conway and pseudo-Conway polynomials#
AUTHORS:
David Roe
Jean-Pierre Flori
Peter Bruin
- class sage.rings.finite_rings.conway_polynomials.PseudoConwayLattice(p, use_database=True)#
Bases:
WithEqualityById
,SageObject
A pseudo-Conway lattice over a given finite prime field.
The Conway polynomial \(f_n\) of degree \(n\) over \(\Bold{F}_p\) is defined by the following four conditions:
\(f_n\) is irreducible.
In the quotient field \(\Bold{F}_p[x]/(f_n)\), the element \(x\bmod f_n\) generates the multiplicative group.
The minimal polynomial of \((x\bmod f_n)^{\frac{p^n-1}{p^m-1}}\) equals the Conway polynomial \(f_m\), for every divisor \(m\) of \(n\).
\(f_n\) is lexicographically least among all such polynomials, under a certain ordering.
The final condition is needed only in order to make the Conway polynomial unique. We define a pseudo-Conway lattice to be any family of polynomials, indexed by the positive integers, satisfying the first three conditions.
INPUT:
p
– prime numberuse_database
– boolean. IfTrue
, use actual Conway polynomials whenever they are available in the database. IfFalse
, always compute pseudo-Conway polynomials.
EXAMPLES:
sage: # needs sage.rings.finite_rings sage: from sage.rings.finite_rings.conway_polynomials import PseudoConwayLattice sage: PCL = PseudoConwayLattice(2, use_database=False) sage: PCL.polynomial(3) x^3 + x + 1
- check_consistency(n)#
Check that the pseudo-Conway polynomials of degree dividing \(n\) in this lattice satisfy the required compatibility conditions.
EXAMPLES:
sage: # needs sage.rings.finite_rings sage: from sage.rings.finite_rings.conway_polynomials import PseudoConwayLattice sage: PCL = PseudoConwayLattice(2, use_database=False) sage: PCL.check_consistency(6) sage: PCL.check_consistency(60) # long time
- polynomial(n)#
Return the pseudo-Conway polynomial of degree \(n\) in this lattice.
INPUT:
n
– positive integer
OUTPUT:
a pseudo-Conway polynomial of degree \(n\) for the prime \(p\).
ALGORITHM:
Uses an algorithm described in [HL1999], modified to find pseudo-Conway polynomials rather than Conway polynomials. The major difference is that we stop as soon as we find a primitive polynomial.
EXAMPLES:
sage: # needs sage.rings.finite_rings sage: from sage.rings.finite_rings.conway_polynomials import PseudoConwayLattice sage: PCL = PseudoConwayLattice(2, use_database=False) sage: PCL.polynomial(3) x^3 + x + 1 sage: PCL.polynomial(4) x^4 + x^3 + 1 sage: PCL.polynomial(60) x^60 + x^59 + x^58 + x^55 + x^54 + x^53 + x^52 + x^51 + x^48 + x^46 + x^45 + x^42 + x^41 + x^39 + x^38 + x^37 + x^35 + x^32 + x^31 + x^30 + x^28 + x^24 + x^22 + x^21 + x^18 + x^17 + x^16 + x^15 + x^14 + x^10 + x^8 + x^7 + x^5 + x^3 + x^2 + x + 1
- sage.rings.finite_rings.conway_polynomials.conway_polynomial(p, n)#
Return the Conway polynomial of degree \(n\) over
GF(p)
.If the requested polynomial is not known, this function raises a
RuntimeError
exception.INPUT:
p
– prime numbern
– positive integer
OUTPUT:
the Conway polynomial of degree \(n\) over the finite field
GF(p)
, loaded from a table.
Note
The first time this function is called a table is read from disk, which takes a fraction of a second. Subsequent calls do not require reloading the table.
See also the
ConwayPolynomials()
object, which is the table of Conway polynomials used by this function.EXAMPLES:
sage: conway_polynomial(2,5) # needs conway_polynomials x^5 + x^2 + 1 sage: conway_polynomial(101,5) # needs conway_polynomials x^5 + 2*x + 99 sage: conway_polynomial(97,101) # needs conway_polynomials Traceback (most recent call last): ... RuntimeError: requested Conway polynomial not in database.
- sage.rings.finite_rings.conway_polynomials.exists_conway_polynomial(p, n)#
Check whether the Conway polynomial of degree \(n\) over
GF(p)
is known.INPUT:
p
– prime numbern
– positive integer
OUTPUT:
boolean:
True
if the Conway polynomial of degree \(n\) overGF(p)
is in the database,False
otherwise.
If the Conway polynomial is in the database, it can be obtained using the command
conway_polynomial(p,n)
.EXAMPLES:
sage: exists_conway_polynomial(2,3) # needs conway_polynomials True sage: exists_conway_polynomial(2,-1) False sage: exists_conway_polynomial(97,200) False sage: exists_conway_polynomial(6,6) False