Splitting fields of polynomials over number fields¶

AUTHORS:

Jeroen Demeyer (2014-01-02): initial version for trac ticket #2217
Jeroen Demeyer (2014-01-03): add abort_degree argument, trac ticket #15626

class sage.rings.number_field.splitting_field.SplittingData(_pol, _dm)¶

Bases: object

A class to store data for internal use in splitting_field(). It contains two attributes pol (polynomial), dm (degree multiple), where pol is a PARI polynomial and dm a Sage Integer.

dm is a multiple of the degree of the splitting field of pol over some field \(E\). In splitting_field(), \(E\) is the field containing the current field \(K\) and all roots of other polynomials inside the list \(L\) with dm less than this dm.

key()¶

Return a sorting key. Compare first by degree bound, then by polynomial degree, then by discriminant.

EXAMPLES:

sage: from sage.rings.number_field.splitting_field import SplittingData
sage: L = []
sage: L.append(SplittingData(pari("x^2 + 1"), 1))
sage: L.append(SplittingData(pari("x^3 + 1"), 1))
sage: L.append(SplittingData(pari("x^2 + 7"), 2))
sage: L.append(SplittingData(pari("x^3 + 1"), 2))
sage: L.append(SplittingData(pari("x^3 + x^2 + x + 1"), 2))
sage: L.sort(key=lambda x: x.key()); L
[SplittingData(x^2 + 1, 1), SplittingData(x^3 + 1, 1), SplittingData(x^2 + 7, 2), SplittingData(x^3 + x^2 + x + 1, 2), SplittingData(x^3 + 1, 2)]
sage: [x.key() for x in L]
[(1, 2, 16), (1, 3, 729), (2, 2, 784), (2, 3, 256), (2, 3, 729)]

poldegree()¶

Return the degree of self.pol

EXAMPLES:

sage: from sage.rings.number_field.splitting_field import SplittingData
sage: SplittingData(pari("x^123 + x + 1"), 2).poldegree()
123

exception sage.rings.number_field.splitting_field.SplittingFieldAbort(div, mult)¶

Bases: Exception

Special exception class to indicate an early abort of splitting_field().

EXAMPLES:

sage: from sage.rings.number_field.splitting_field import SplittingFieldAbort
sage: raise SplittingFieldAbort(20, 60)
Traceback (most recent call last):
...
SplittingFieldAbort: degree of splitting field is a multiple of 20
sage: raise SplittingFieldAbort(12, 12)
Traceback (most recent call last):
...
SplittingFieldAbort: degree of splitting field equals 12

sage.rings.number_field.splitting_field.splitting_field(poly, name, map=False, degree_multiple=None, abort_degree=None, simplify=True, simplify_all=False)¶

Compute the splitting field of a given polynomial, defined over a number field.

INPUT:

poly – a monic polynomial over a number field
name – a variable name for the number field
map – (default: False) also return an embedding of poly into the resulting field. Note that computing this embedding might be expensive.
degree_multiple – a multiple of the absolute degree of the splitting field. If degree_multiple equals the actual degree, this can enormously speed up the computation.
abort_degree – abort by raising a SplittingFieldAbort if it can be determined that the absolute degree of the splitting field is strictly larger than abort_degree.
simplify – (default: True) during the algorithm, try to find a simpler defining polynomial for the intermediate number fields using PARI’s polred(). This usually speeds up the computation but can also considerably slow it down. Try and see what works best in the given situation.
simplify_all – (default: False) If True, simplify intermediate fields and also the resulting number field.

OUTPUT:

If map is False, the splitting field as an absolute number field. If map is True, a tuple (K, phi) where phi is an embedding of the base field in K.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = (x^3 + 2).splitting_field(); K
Number Field in a with defining polynomial x^6 + 3*x^5 + 6*x^4 + 11*x^3 + 12*x^2 - 3*x + 1
sage: K.<a> = (x^3 - 3*x + 1).splitting_field(); K
Number Field in a with defining polynomial x^3 - 3*x + 1

The simplify and simplify_all flags usually yield fields defined by polynomials with smaller coefficients. By default, simplify is True and simplify_all is False.

sage: (x^4 - x + 1).splitting_field('a', simplify=False)
Number Field in a with defining polynomial x^24 - 2780*x^22 + 2*x^21 + 3527512*x^20 - 2876*x^19 - 2701391985*x^18 + 945948*x^17 + 1390511639677*x^16 + 736757420*x^15 - 506816498313560*x^14 - 822702898220*x^13 + 134120588299548463*x^12 + 362240696528256*x^11 - 25964582366880639486*x^10 - 91743672243419990*x^9 + 3649429473447308439427*x^8 + 14310332927134072336*x^7 - 363192569823568746892571*x^6 - 1353403793640477725898*x^5 + 24293393281774560140427565*x^4 + 70673814899934142357628*x^3 - 980621447508959243128437933*x^2 - 1539841440617805445432660*x + 18065914012013502602456565991
sage: (x^4 - x + 1).splitting_field('a', simplify=True)
Number Field in a with defining polynomial x^24 + 8*x^23 - 32*x^22 - 310*x^21 + 540*x^20 + 4688*x^19 - 6813*x^18 - 32380*x^17 + 49525*x^16 + 102460*x^15 - 129944*x^14 - 287884*x^13 + 372727*x^12 + 150624*x^11 - 110530*x^10 - 566926*x^9 + 1062759*x^8 - 779940*x^7 + 863493*x^6 - 1623578*x^5 + 1759513*x^4 - 955624*x^3 + 459975*x^2 - 141948*x + 53919
sage: (x^4 - x + 1).splitting_field('a', simplify_all=True)
Number Field in a with defining polynomial x^24 - 3*x^23 + 2*x^22 - x^20 + 4*x^19 + 32*x^18 - 35*x^17 - 92*x^16 + 49*x^15 + 163*x^14 - 15*x^13 - 194*x^12 - 15*x^11 + 163*x^10 + 49*x^9 - 92*x^8 - 35*x^7 + 32*x^6 + 4*x^5 - x^4 + 2*x^2 - 3*x + 1

Reducible polynomials also work:

sage: pol = (x^4 - 1)*(x^2 + 1/2)*(x^2 + 1/3)
sage: pol.splitting_field('a', simplify_all=True)
Number Field in a with defining polynomial x^8 - x^4 + 1

Relative situation:

sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(x^3 + 2)
sage: S.<t> = PolynomialRing(K)
sage: L.<b> = (t^2 - a).splitting_field()
sage: L
Number Field in b with defining polynomial t^6 + 2

With map=True, we also get the embedding of the base field into the splitting field:

sage: L.<b>, phi = (t^2 - a).splitting_field(map=True)
sage: phi
Ring morphism:
  From: Number Field in a with defining polynomial x^3 + 2
  To:   Number Field in b with defining polynomial t^6 + 2
  Defn: a |--> b^2
sage: (x^4 - x + 1).splitting_field('a', simplify_all=True, map=True)[1]
Ring morphism:
  From: Rational Field
  To:   Number Field in a with defining polynomial x^24 - 3*x^23 + 2*x^22 - x^20 + 4*x^19 + 32*x^18 - 35*x^17 - 92*x^16 + 49*x^15 + 163*x^14 - 15*x^13 - 194*x^12 - 15*x^11 + 163*x^10 + 49*x^9 - 92*x^8 - 35*x^7 + 32*x^6 + 4*x^5 - x^4 + 2*x^2 - 3*x + 1
  Defn: 1 |--> 1

We can enable verbose messages:

sage: from sage.misc.verbose import set_verbose
sage: set_verbose(2)
sage: K.<a> = (x^3 - x + 1).splitting_field()
verbose 1 (...: splitting_field.py, splitting_field) Starting field: y
verbose 1 (...: splitting_field.py, splitting_field) SplittingData to factor: [(3, 0)]
verbose 2 (...: splitting_field.py, splitting_field) Done factoring (time = ...)
verbose 1 (...: splitting_field.py, splitting_field) SplittingData to handle: [(2, 2), (3, 3)]
verbose 1 (...: splitting_field.py, splitting_field) Bounds for absolute degree: [6, 6]
verbose 2 (...: splitting_field.py, splitting_field) Handling polynomial x^2 + 23
verbose 1 (...: splitting_field.py, splitting_field) New field before simplifying: x^2 + 23 (time = ...)
verbose 1 (...: splitting_field.py, splitting_field) New field: y^2 - y + 6 (time = ...)
verbose 2 (...: splitting_field.py, splitting_field) Converted polynomials to new field (time = ...)
verbose 1 (...: splitting_field.py, splitting_field) SplittingData to factor: []
verbose 2 (...: splitting_field.py, splitting_field) Done factoring (time = ...)
verbose 1 (...: splitting_field.py, splitting_field) SplittingData to handle: [(3, 3)]
verbose 1 (...: splitting_field.py, splitting_field) Bounds for absolute degree: [6, 6]
verbose 2 (...: splitting_field.py, splitting_field) Handling polynomial x^3 - x + 1
verbose 1 (...: splitting_field.py, splitting_field) New field: y^6 + 3*y^5 + 19*y^4 + 35*y^3 + 127*y^2 + 73*y + 271 (time = ...)
sage: set_verbose(0)

Try all Galois groups in degree 4. We use a quadratic base field such that polgalois() cannot be used:

sage: R.<x> = PolynomialRing(QuadraticField(-11))
sage: C2C2pol = x^4 - 10*x^2 + 1
sage: C2C2pol.splitting_field('x')
Number Field in x with defining polynomial x^8 + 24*x^6 + 608*x^4 + 9792*x^2 + 53824
sage: C4pol = x^4 + x^3 + x^2 + x + 1
sage: C4pol.splitting_field('x')
Number Field in x with defining polynomial x^8 - x^7 - 2*x^6 + 5*x^5 + x^4 + 15*x^3 - 18*x^2 - 27*x + 81
sage: D8pol = x^4 - 2
sage: D8pol.splitting_field('x')
Number Field in x with defining polynomial x^16 + 8*x^15 + 68*x^14 + 336*x^13 + 1514*x^12 + 5080*x^11 + 14912*x^10 + 35048*x^9 + 64959*x^8 + 93416*x^7 + 88216*x^6 + 41608*x^5 - 25586*x^4 - 60048*x^3 - 16628*x^2 + 12008*x + 34961
sage: A4pol = x^4 - 4*x^3 + 14*x^2 - 28*x + 21
sage: A4pol.splitting_field('x')
Number Field in x with defining polynomial x^24 - 20*x^23 + 290*x^22 - 3048*x^21 + 26147*x^20 - 186132*x^19 + 1130626*x^18 - 5913784*x^17 + 26899345*x^16 - 106792132*x^15 + 371066538*x^14 - 1127792656*x^13 + 2991524876*x^12 - 6888328132*x^11 + 13655960064*x^10 - 23000783036*x^9 + 32244796382*x^8 - 36347834476*x^7 + 30850889884*x^6 - 16707053128*x^5 + 1896946429*x^4 + 4832907884*x^3 - 3038258802*x^2 - 200383596*x + 593179173
sage: S4pol = x^4 + x + 1
sage: S4pol.splitting_field('x')
Number Field in x with defining polynomial x^48 ...

Some bigger examples:

sage: R.<x> = PolynomialRing(QQ)
sage: pol15 = chebyshev_T(31, x) - 1    # 2^30*(x-1)*minpoly(cos(2*pi/31))^2
sage: pol15.splitting_field('a')
Number Field in a with defining polynomial x^15 - x^14 - 14*x^13 + 13*x^12 + 78*x^11 - 66*x^10 - 220*x^9 + 165*x^8 + 330*x^7 - 210*x^6 - 252*x^5 + 126*x^4 + 84*x^3 - 28*x^2 - 8*x + 1
sage: pol48 = x^6 - 4*x^4 + 12*x^2 - 12
sage: pol48.splitting_field('a')
Number Field in a with defining polynomial x^48 ...

If you somehow know the degree of the field in advance, you should add a degree_multiple argument. This can speed up the computation, in particular for polynomials of degree >= 12 or for relative extensions:

sage: pol15.splitting_field('a', degree_multiple=15)
Number Field in a with defining polynomial x^15 + x^14 - 14*x^13 - 13*x^12 + 78*x^11 + 66*x^10 - 220*x^9 - 165*x^8 + 330*x^7 + 210*x^6 - 252*x^5 - 126*x^4 + 84*x^3 + 28*x^2 - 8*x - 1

A value for degree_multiple which isn’t actually a multiple of the absolute degree of the splitting field can either result in a wrong answer or the following exception:

sage: pol48.splitting_field('a', degree_multiple=20)
Traceback (most recent call last):
...
ValueError: inconsistent degree_multiple in splitting_field()

Compute the Galois closure as the splitting field of the defining polynomial:

sage: R.<x> = PolynomialRing(QQ)
sage: pol48 = x^6 - 4*x^4 + 12*x^2 - 12
sage: K.<a> = NumberField(pol48)
sage: L.<b> = pol48.change_ring(K).splitting_field()
sage: L
Number Field in b with defining polynomial x^48 ...

Try all Galois groups over \(\QQ\) in degree 5 except for \(S_5\) (the latter is infeasible with the current implementation):

sage: C5pol = x^5 + x^4 - 4*x^3 - 3*x^2 + 3*x + 1
sage: C5pol.splitting_field('x')
Number Field in x with defining polynomial x^5 + x^4 - 4*x^3 - 3*x^2 + 3*x + 1
sage: D10pol = x^5 - x^4 - 5*x^3 + 4*x^2 + 3*x - 1
sage: D10pol.splitting_field('x')
Number Field in x with defining polynomial x^10 - 28*x^8 + 216*x^6 - 681*x^4 + 902*x^2 - 401
sage: AGL_1_5pol = x^5 - 2
sage: AGL_1_5pol.splitting_field('x')
Number Field in x with defining polynomial x^20 + 10*x^19 + 55*x^18 + 210*x^17 + 595*x^16 + 1300*x^15 + 2250*x^14 + 3130*x^13 + 3585*x^12 + 3500*x^11 + 2965*x^10 + 2250*x^9 + 1625*x^8 + 1150*x^7 + 750*x^6 + 400*x^5 + 275*x^4 + 100*x^3 + 75*x^2 + 25
sage: A5pol = x^5 - x^4 + 2*x^2 - 2*x + 2
sage: A5pol.splitting_field('x')
Number Field in x with defining polynomial x^60 ...

We can use the abort_degree option if we don’t want to compute fields of too large degree (this can be used to check whether the splitting field has small degree):

sage: (x^5+x+3).splitting_field('b', abort_degree=119)
Traceback (most recent call last):
...
SplittingFieldAbort: degree of splitting field equals 120
sage: (x^10+x+3).splitting_field('b', abort_degree=60)  # long time (10s on sage.math, 2014)
Traceback (most recent call last):
...
SplittingFieldAbort: degree of splitting field is a multiple of 180

Use the degree_divisor attribute to recover the divisor of the degree of the splitting field or degree_multiple to recover a multiple:

sage: from sage.rings.number_field.splitting_field import SplittingFieldAbort
sage: try:  # long time (4s on sage.math, 2014)
....:     (x^8+x+1).splitting_field('b', abort_degree=60, simplify=False)
....: except SplittingFieldAbort as e:
....:     print(e.degree_divisor)
....:     print(e.degree_multiple)
120
1440

Splitting fields of polynomials over number fields¶

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