Coverage for local/lib/python2.7/site-packages/sage/rings/number_field/galois_group.py : 91%
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""" Galois Groups of Number Fields
AUTHORS:
- William Stein (2004, 2005): initial version - David Loeffler (2009): rewrite to give explicit homomorphism groups
TESTS:
Standard test of pickleability::
sage: G = NumberField(x^3 + 2, 'alpha').galois_group(type="pari"); G Galois group PARI group [6, -1, 2, "S3"] of degree 3 of the Number Field in alpha with defining polynomial x^3 + 2 sage: G == loads(dumps(G)) True
sage: G = NumberField(x^3 + 2, 'alpha').galois_group(names='beta'); G Galois group of Galois closure in beta of Number Field in alpha with defining polynomial x^3 + 2 sage: G == loads(dumps(G)) True """
r""" A wrapper around a class representing an abstract transitive group.
This is just a fairly minimal object at present. To get the underlying group, do ``G.group()``, and to get the corresponding number field do ``G.number_field()``. For a more sophisticated interface use the ``type=None`` option.
EXAMPLES::
sage: K = QQ[2^(1/3)] sage: G = K.galois_group(type="pari"); G Galois group PARI group [6, -1, 2, "S3"] of degree 3 of the Number Field in a with defining polynomial x^3 - 2 sage: G.order() 6 sage: G.group() PARI group [6, -1, 2, "S3"] of degree 3 sage: G.number_field() Number Field in a with defining polynomial x^3 - 2 """
""" Create a Galois group.
EXAMPLES::
sage: NumberField([x^2 + 1, x^2 + 2],'a').galois_group(type="pari") Galois group PARI group [4, 1, 2, "E(4) = 2[x]2"] of degree 4 of the Number Field in a0 with defining polynomial x^2 + 1 over its base field """
""" Compare two number field Galois groups.
First the number fields are compared, then the Galois groups if the number fields are equal. (Of course, if the number fields are the same, the Galois groups are automatically equal.)
EXAMPLES::
sage: G = NumberField(x^3 + 2, 'alpha').galois_group(type="pari") sage: H = QQ[sqrt(2)].galois_group(type="pari") sage: H == G False sage: H == H True sage: G == G True """ return False return True
""" Test for unequality.
EXAMPLES::
sage: G = NumberField(x^3 + 2, 'alpha').galois_group(type="pari") sage: H = QQ[sqrt(2)].galois_group(type="pari") sage: H != G True sage: H != H False sage: G != G False """
""" Display print representation of a Galois group.
EXAMPLES::
sage: G = NumberField(x^4 + 2*x + 2, 'a').galois_group(type="pari") sage: G.__repr__() 'Galois group PARI group [24, -1, 5, "S4"] of degree 4 of the Number Field in a with defining polynomial x^4 + 2*x + 2' """ self.__number_field)
""" Return the underlying abstract group.
EXAMPLES::
sage: G = NumberField(x^3 + 2*x + 2, 'theta').galois_group(type="pari") sage: H = G.group(); H PARI group [6, -1, 2, "S3"] of degree 3 sage: P = H.permutation_group(); P # optional - database_gap Transitive group number 2 of degree 3 sage: list(P) # optional - database_gap [(), (1,2), (1,2,3), (2,3), (1,3,2), (1,3)] """
""" Return the order of this Galois group.
EXAMPLES::
sage: G = NumberField(x^5 + 2, 'theta_1').galois_group(type="pari"); G Galois group PARI group [20, -1, 3, "F(5) = 5:4"] of degree 5 of the Number Field in theta_1 with defining polynomial x^5 + 2 sage: G.order() 20 """
""" Return the number field of which this is the Galois group.
EXAMPLES::
sage: G = NumberField(x^6 + 2, 't').galois_group(type="pari"); G Galois group PARI group [12, -1, 3, "D(6) = S(3)[x]2"] of degree 6 of the Number Field in t with defining polynomial x^6 + 2 sage: G.number_field() Number Field in t with defining polynomial x^6 + 2 """
r""" The Galois group of an (absolute) number field.
.. note::
We define the Galois group of a non-normal field K to be the Galois group of its Galois closure L, and elements are stored as permutations of the roots of the defining polynomial of L, *not* as permutations of the roots (in L) of the defining polynomial of K. The latter would probably be preferable, but is harder to implement. Thus the permutation group that is returned is always simply-transitive.
The 'arithmetical' features (decomposition and ramification groups, Artin symbols etc) are only available for Galois fields. """
r""" Create a Galois group.
EXAMPLES::
sage: QuadraticField(-23,'a').galois_group() Galois group of Number Field in a with defining polynomial x^2 + 23 sage: NumberField(x^3 - 2, 'b').galois_group() Traceback (most recent call last): ... TypeError: You must specify the name of the generator. sage: NumberField(x^3 - 2, 'b').galois_group(names="c") Galois group of Galois closure in c of Number Field in b with defining polynomial x^3 - 2 """
else:
# Sort the vector of permutations using .list() as key to avoid errors # from using comparison operators on non-scalar PARI objects. sorted(g[6], key=lambda x: x.list()))
# PARI computes all the elements of self anyway, so we might as well store them
r""" Return the class to be used for creating elements of this group, which is GaloisGroupElement.
EXAMPLES::
sage: F.<z> = CyclotomicField(7) sage: G = F.galois_group() sage: G._element_class() <class 'sage.rings.number_field.galois_group.GaloisGroupElement'>
We test that a method inherited from PermutationGroup_generic returns the right type of element (see :trac:`133`)::
sage: phi = G.random_element() sage: type(phi) <class 'sage.rings.number_field.galois_group.GaloisGroupElement'> sage: phi(z) # random z^3 """
r""" Create an element of self from x. Here x had better be one of: -- the integer 1, denoting the identity of G -- an element of G -- a permutation of the right length which defines an element of G, or anything that coerces into a permutation of the right length -- an abstract automorphism of the underlying number field.
EXAMPLES::
sage: K.<a> = QuadraticField(-23) sage: G = K.galois_group() sage: G(1) () sage: G(G.gens()[0]) (1,2) sage: G([(1,2)]) (1,2) sage: G(K.hom(-a, K)) (1,2) """
raise ArithmeticError
r""" Return True if the underlying number field of self is actually Galois.
EXAMPLES::
sage: NumberField(x^3 - x + 1,'a').galois_group(names='b').is_galois() False sage: NumberField(x^2 - x + 1,'a').galois_group().is_galois() True """ else:
r""" Number of generators of self.
EXAMPLES::
sage: QuadraticField(-23, 'a').galois_group().ngens() 1 """
r""" String representation of self.
EXAMPLES::
sage: G = QuadraticField(-23, 'a').galois_group() sage: G._repr_() 'Galois group of Number Field in a with defining polynomial x^2 + 23' sage: G = NumberField(x^3 - 2, 'a').galois_group(names='b') sage: G._repr_() 'Galois group of Galois closure in b of Number Field in a with defining polynomial x^3 - 2' """ else:
r""" The ambient number field.
EXAMPLES::
sage: K = NumberField(x^3 - x + 1, 'a') sage: K.galois_group(names='b').number_field() is K True """
r""" The Galois closure of the ambient number field.
EXAMPLES::
sage: K = NumberField(x^3 - x + 1, 'a') sage: K.galois_group(names='b').splitting_field() Number Field in b with defining polynomial x^6 - 6*x^4 + 9*x^2 + 23 sage: L = QuadraticField(-23, 'c'); L.galois_group().splitting_field() is L True """
r""" List of the elements of self.
EXAMPLES::
sage: NumberField(x^3 - 3*x + 1,'a').galois_group().list() [(), (1,2,3), (1,3,2)] """
r""" Return the ``i``-th element of ``self``.
INPUT:
- ``i`` -- integer between ``0`` and ``n-1`` where ``n`` is the cardinality of this set
EXAMPLES::
sage: G = NumberField(x^3 - 3*x + 1,'a').galois_group() sage: [G.unrank(i) for i in range(G.cardinality())] [(), (1,2,3), (1,3,2)]
TESTS::
sage: G = NumberField(x^3 - 3*x + 1,'a').galois_group() sage: L = [G.unrank(i) for i in range(G.cardinality())] sage: L == G.list() True """
""" Iterate over ``self``.
EXAMPLES::
sage: G = NumberField(x^3 - 3*x + 1,'a').galois_group() sage: list(G) == G.list() True """
r""" Return the subgroup of self with the given elements. Mostly for internal use.
EXAMPLES::
sage: G = NumberField(x^3 - x - 1, 'a').galois_closure('b').galois_group() sage: G.subgroup([ G(1), G([(1,2,3),(4,5,6)]), G([(1,3,2),(4,6,5)]) ]) Subgroup [(), (1,2,3)(4,5,6), (1,3,2)(4,6,5)] of Galois group of Number Field in b with defining polynomial x^6 - 6*x^4 + 9*x^2 + 23 """ else:
# Proper number theory starts here. All the functions below make no sense # unless the field is Galois.
""" Decomposition group of a prime ideal P, i.e. the subgroup of elements that map P to itself. This is the same as the Galois group of the extension of local fields obtained by completing at P.
This function will raise an error if P is not prime or the given number field is not Galois.
P can also be an infinite prime, i.e. an embedding into `\RR` or `\CC`.
EXAMPLES::
sage: K.<a> = NumberField(x^4 - 2*x^2 + 2,'b').galois_closure() sage: P = K.ideal([17, a^2]) sage: G = K.galois_group() sage: G.decomposition_group(P) Subgroup [(), (1,8)(2,7)(3,6)(4,5)] of Galois group of Number Field in a with defining polynomial x^8 - 20*x^6 + 104*x^4 - 40*x^2 + 1156 sage: G.decomposition_group(P^2) Traceback (most recent call last): ... ValueError: Fractional ideal (...) is not prime sage: G.decomposition_group(17) Traceback (most recent call last): ... ValueError: Fractional ideal (17) is not prime
An example with an infinite place::
sage: L.<b> = NumberField(x^3 - 2,'a').galois_closure(); G=L.galois_group() sage: x = L.places()[0] sage: G.decomposition_group(x).order() 2 """ raise TypeError("Decomposition groups only defined for Galois extensions")
return self.subgroup([self.identity()]) else: else:
""" Return the unique element of self corresponding to complex conjugation, for a specified embedding P into the complex numbers. If P is not specified, use the "standard" embedding, whenever that is well-defined.
EXAMPLES::
sage: L.<z> = CyclotomicField(7) sage: G = L.galois_group() sage: conj = G.complex_conjugation(); conj (1,4)(2,5)(3,6) sage: conj(z) -z^5 - z^4 - z^3 - z^2 - z - 1
An example where the field is not CM, so complex conjugation really depends on the choice of embedding::
sage: L = NumberField(x^6 + 40*x^3 + 1372,'a') sage: G = L.galois_group() sage: [G.complex_conjugation(x) for x in L.places()] [(1,3)(2,6)(4,5), (1,5)(2,4)(3,6), (1,2)(3,4)(5,6)] """ raise ValueError("No default complex embedding specified")
raise TypeError("Extension is not Galois") raise TypeError("No complex conjugation (field is real)")
raise ArithmeticError("Something has gone very wrong here")
""" Return the vth ramification group of self for the prime P, i.e. the set of elements s of self such that s acts trivially modulo P^(v+1). This is only defined for Galois fields.
EXAMPLES::
sage: K.<b> = NumberField(x^3 - 3,'a').galois_closure() sage: G=K.galois_group() sage: P = K.primes_above(3)[0] sage: G.ramification_group(P, 3) Subgroup [(), (1,2,4)(3,5,6), (1,4,2)(3,6,5)] of Galois group of Number Field in b with defining polynomial x^6 + 243 sage: G.ramification_group(P, 5) Subgroup [()] of Galois group of Number Field in b with defining polynomial x^6 + 243 """ raise TypeError("Ramification groups only defined for Galois extensions") raise ValueError("%s is not prime")
""" Return the inertia group of the prime P, i.e. the group of elements acting trivially modulo P. This is just the 0th ramification group of P.
EXAMPLES::
sage: K.<b> = NumberField(x^2 - 3,'a') sage: G = K.galois_group() sage: G.inertia_group(K.primes_above(2)[0]) Galois group of Number Field in b with defining polynomial x^2 - 3 sage: G.inertia_group(K.primes_above(5)[0]) Subgroup [()] of Galois group of Number Field in b with defining polynomial x^2 - 3 """ raise TypeError("Inertia groups only defined for Galois extensions")
r""" Return the set of ramification breaks of the prime ideal P, i.e. the set of indices i such that the ramification group `G_{i+1} \ne G_{i}`. This is only defined for Galois fields.
EXAMPLES::
sage: K.<b> = NumberField(x^8 - 20*x^6 + 104*x^4 - 40*x^2 + 1156) sage: G = K.galois_group() sage: P = K.primes_above(2)[0] sage: G.ramification_breaks(P) {1, 3, 5} sage: min( [ G.ramification_group(P, i).order() / G.ramification_group(P, i+1).order() for i in G.ramification_breaks(P)] ) 2 """ raise TypeError("Ramification breaks only defined for Galois extensions")
r""" Return the Artin symbol `\left(\frac{K / \QQ}{\mathfrak{P}}\right)`, where K is the number field of self, and `\mathfrak{P}` is an unramified prime ideal. This is the unique element s of the decomposition group of `\mathfrak{P}` such that `s(x) = x^p \bmod \mathfrak{P}`, where p is the residue characteristic of `\mathfrak{P}`.
EXAMPLES::
sage: K.<b> = NumberField(x^4 - 2*x^2 + 2, 'a').galois_closure() sage: G = K.galois_group() sage: [G.artin_symbol(P) for P in K.primes_above(7)] [(1,5)(2,6)(3,7)(4,8), (1,5)(2,6)(3,7)(4,8), (1,4)(2,3)(5,8)(6,7), (1,4)(2,3)(5,8)(6,7)] sage: G.artin_symbol(17) Traceback (most recent call last): ... ValueError: Fractional ideal (17) is not prime sage: QuadraticField(-7,'c').galois_group().artin_symbol(13) (1,2) sage: G.artin_symbol(K.primes_above(2)[0]) Traceback (most recent call last): ... ValueError: Fractional ideal (...) is ramified """ raise TypeError("Artin symbols only defined for Galois extensions")
r""" A subgroup of a Galois group, as returned by functions such as ``decomposition_group``. """
r""" Create a subgroup of a Galois group with the given elements. It is generally better to use the subgroup() method of the parent group.
EXAMPLES::
sage: from sage.rings.number_field.galois_group import GaloisGroup_subgroup sage: G = NumberField(x^3 - x - 1, 'a').galois_closure('b').galois_group() sage: GaloisGroup_subgroup( G, [ G(1), G([(1,2,3),(4,5,6)]), G([(1,3,2),(4,6,5)])]) Subgroup [(), (1,2,3)(4,5,6), (1,3,2)(4,6,5)] of Galois group of Number Field in b with defining polynomial x^6 - 6*x^4 + 9*x^2 + 23
TESTS:
Check that :trac:`17664` is fixed::
sage: L.<c> = QuadraticField(-1) sage: P = L.primes_above(5)[0] sage: G = L.galois_group() sage: H = G.decomposition_group(P) sage: H.domain() {1, 2} sage: G.artin_symbol(P) () """ # XXX This should be fixed so that this can use GaloisGroup_v2.__init__ domain=ambient.domain())
r""" Return the fixed field of this subgroup (as a subfield of the Galois closure of the number field associated to the ambient Galois group).
EXAMPLES::
sage: L.<a> = NumberField(x^4 + 1) sage: G = L.galois_group() sage: H = G.decomposition_group(L.primes_above(3)[0]) sage: H.fixed_field() (Number Field in a0 with defining polynomial x^2 + 2, Ring morphism: From: Number Field in a0 with defining polynomial x^2 + 2 To: Number Field in a with defining polynomial x^4 + 1 Defn: a0 |--> a^3 + a)
""" return self._galois_closure # work around a silly error
r""" String representation of self.
EXAMPLES::
sage: G = NumberField(x^3 - x - 1, 'a').galois_closure('b').galois_group() sage: H = G.subgroup([ G(1), G([(1,2,3),(4,5,6)]), G([(1,3,2),(4,6,5)])]) sage: H._repr_() 'Subgroup [(), (1,2,3)(4,5,6), (1,3,2)(4,6,5)] of Galois group of Number Field in b with defining polynomial x^6 - 6*x^4 + 9*x^2 + 23' """
r""" An element of a Galois group. This is stored as a permutation, but may also be made to act on elements of the field (generally returning elements of its Galois closure).
EXAMPLES::
sage: K.<w> = QuadraticField(-7); G = K.galois_group() sage: G[1] (1,2) sage: G[1](w + 2) -w + 2
sage: L.<v> = NumberField(x^3 - 2); G = L.galois_group(names='y') sage: G[4] (1,5)(2,4)(3,6) sage: G[4](v) 1/18*y^4 sage: G[4](G[4](v)) -1/36*y^4 - 1/2*y sage: G[4](G[4](G[4](v))) 1/18*y^4 """ def as_hom(self): r""" Return the homomorphism L -> L corresponding to self, where L is the Galois closure of the ambient number field.
EXAMPLES::
sage: G = QuadraticField(-7,'w').galois_group() sage: G[1].as_hom() Ring endomorphism of Number Field in w with defining polynomial x^2 + 7 Defn: w |--> -w
TESTS:
Number fields defined by non-monic and non-integral polynomials are supported (:trac:`252`)::
sage: R.<x> = QQ[] sage: f = 7/9*x^3 + 7/3*x^2 - 56*x + 123 sage: K.<a> = NumberField(f) sage: G = K.galois_group() sage: G[1].as_hom() Ring endomorphism of Number Field in a with defining polynomial 7/9*x^3 + 7/3*x^2 - 56*x + 123 Defn: a |--> -7/15*a^2 - 18/5*a + 96/5 sage: prod(x - sigma(a) for sigma in G) == f.monic() True """ # First compute the image of the standard generator of the # PARI number field. # Now convert this to a conjugate of the standard generator of # the Sage number field.
r""" Return the action of self on an element x in the number field of self (or its Galois closure).
EXAMPLES::
sage: K.<w> = QuadraticField(-7) sage: f = K.galois_group()[1] sage: f(w) -w """ else:
""" Return the greatest value of v such that s acts trivially modulo P^v. Should only be used if P is prime and s is in the decomposition group of P.
EXAMPLES::
sage: K.<b> = NumberField(x^3 - 3, 'a').galois_closure() sage: G = K.galois_group() sage: P = K.primes_above(3)[0] sage: s = hom(K, K, 1/18*b^4 - 1/2*b) sage: G(s).ramification_degree(P) 4 """ raise TypeError("Ramification degree only defined for Galois extensions")
# For unpickling purposes we rebind GaloisGroup as GaloisGroup_v1.
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