144 lines
4.9 KiB
Python
144 lines
4.9 KiB
Python
"""Tests for computing Galois groups. """
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from sympy.abc import x
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from sympy.combinatorics.galois import (
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S1TransitiveSubgroups, S2TransitiveSubgroups, S3TransitiveSubgroups,
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S4TransitiveSubgroups, S5TransitiveSubgroups, S6TransitiveSubgroups,
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)
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from sympy.polys.domains.rationalfield import QQ
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from sympy.polys.numberfields.galoisgroups import (
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tschirnhausen_transformation,
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galois_group,
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_galois_group_degree_4_root_approx,
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_galois_group_degree_5_hybrid,
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)
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from sympy.polys.numberfields.subfield import field_isomorphism
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from sympy.polys.polytools import Poly
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from sympy.testing.pytest import raises
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def test_tschirnhausen_transformation():
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for T in [
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Poly(x**2 - 2),
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Poly(x**2 + x + 1),
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Poly(x**4 + 1),
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Poly(x**4 - x**3 + x**2 - x + 1),
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]:
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_, U = tschirnhausen_transformation(T)
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assert U.degree() == T.degree()
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assert U.is_monic
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assert U.is_irreducible
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K = QQ.alg_field_from_poly(T)
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L = QQ.alg_field_from_poly(U)
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assert field_isomorphism(K.ext, L.ext) is not None
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# Test polys are from:
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# Cohen, H. *A Course in Computational Algebraic Number Theory*.
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test_polys_by_deg = {
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# Degree 1
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1: [
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(x, S1TransitiveSubgroups.S1, True)
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],
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# Degree 2
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2: [
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(x**2 + x + 1, S2TransitiveSubgroups.S2, False)
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],
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# Degree 3
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3: [
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(x**3 + x**2 - 2*x - 1, S3TransitiveSubgroups.A3, True),
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(x**3 + 2, S3TransitiveSubgroups.S3, False),
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],
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# Degree 4
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4: [
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(x**4 + x**3 + x**2 + x + 1, S4TransitiveSubgroups.C4, False),
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(x**4 + 1, S4TransitiveSubgroups.V, True),
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(x**4 - 2, S4TransitiveSubgroups.D4, False),
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(x**4 + 8*x + 12, S4TransitiveSubgroups.A4, True),
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(x**4 + x + 1, S4TransitiveSubgroups.S4, False),
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],
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# Degree 5
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5: [
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(x**5 + x**4 - 4*x**3 - 3*x**2 + 3*x + 1, S5TransitiveSubgroups.C5, True),
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(x**5 - 5*x + 12, S5TransitiveSubgroups.D5, True),
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(x**5 + 2, S5TransitiveSubgroups.M20, False),
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(x**5 + 20*x + 16, S5TransitiveSubgroups.A5, True),
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(x**5 - x + 1, S5TransitiveSubgroups.S5, False),
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],
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# Degree 6
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6: [
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(x**6 + x**5 + x**4 + x**3 + x**2 + x + 1, S6TransitiveSubgroups.C6, False),
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(x**6 + 108, S6TransitiveSubgroups.S3, False),
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(x**6 + 2, S6TransitiveSubgroups.D6, False),
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(x**6 - 3*x**2 - 1, S6TransitiveSubgroups.A4, True),
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(x**6 + 3*x**3 + 3, S6TransitiveSubgroups.G18, False),
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(x**6 - 3*x**2 + 1, S6TransitiveSubgroups.A4xC2, False),
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(x**6 - 4*x**2 - 1, S6TransitiveSubgroups.S4p, True),
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(x**6 - 3*x**5 + 6*x**4 - 7*x**3 + 2*x**2 + x - 4, S6TransitiveSubgroups.S4m, False),
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(x**6 + 2*x**3 - 2, S6TransitiveSubgroups.G36m, False),
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(x**6 + 2*x**2 + 2, S6TransitiveSubgroups.S4xC2, False),
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(x**6 + 10*x**5 + 55*x**4 + 140*x**3 + 175*x**2 + 170*x + 25, S6TransitiveSubgroups.PSL2F5, True),
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(x**6 + 10*x**5 + 55*x**4 + 140*x**3 + 175*x**2 - 3019*x + 25, S6TransitiveSubgroups.PGL2F5, False),
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(x**6 + 6*x**4 + 2*x**3 + 9*x**2 + 6*x - 4, S6TransitiveSubgroups.G36p, True),
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(x**6 + 2*x**4 + 2*x**3 + x**2 + 2*x + 2, S6TransitiveSubgroups.G72, False),
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(x**6 + 24*x - 20, S6TransitiveSubgroups.A6, True),
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(x**6 + x + 1, S6TransitiveSubgroups.S6, False),
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],
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}
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def test_galois_group():
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"""
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Try all the test polys.
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"""
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for deg in range(1, 7):
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polys = test_polys_by_deg[deg]
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for T, G, alt in polys:
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assert galois_group(T, by_name=True) == (G, alt)
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def test_galois_group_degree_out_of_bounds():
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raises(ValueError, lambda: galois_group(Poly(0, x)))
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raises(ValueError, lambda: galois_group(Poly(1, x)))
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raises(ValueError, lambda: galois_group(Poly(x ** 7 + 1)))
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def test_galois_group_not_by_name():
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"""
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Check at least one polynomial of each supported degree, to see that
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conversion from name to group works.
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"""
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for deg in range(1, 7):
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T, G_name, _ = test_polys_by_deg[deg][0]
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G, _ = galois_group(T)
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assert G == G_name.get_perm_group()
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def test_galois_group_not_monic_over_ZZ():
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"""
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Check that we can work with polys that are not monic over ZZ.
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"""
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for deg in range(1, 7):
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T, G, alt = test_polys_by_deg[deg][0]
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assert galois_group(T/2, by_name=True) == (G, alt)
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def test__galois_group_degree_4_root_approx():
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for T, G, alt in test_polys_by_deg[4]:
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assert _galois_group_degree_4_root_approx(Poly(T)) == (G, alt)
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def test__galois_group_degree_5_hybrid():
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for T, G, alt in test_polys_by_deg[5]:
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assert _galois_group_degree_5_hybrid(Poly(T)) == (G, alt)
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def test_AlgebraicField_galois_group():
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k = QQ.alg_field_from_poly(Poly(x**4 + 1))
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G, _ = k.galois_group(by_name=True)
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assert G == S4TransitiveSubgroups.V
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k = QQ.alg_field_from_poly(Poly(x**4 - 2))
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G, _ = k.galois_group(by_name=True)
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assert G == S4TransitiveSubgroups.D4
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