# HG changeset patch # User Craig Citro # Date 1261466849 28800 # Node ID faaaf3d2cbae961e7e90eee7434165bbcb98c949 # Parent f872a92f5f9afff4b4971afc82bf9270faec269b Fix trac #7616 -- bug in code for relative number fields of degree one. diff -r f872a92f5f9a -r faaaf3d2cbae sage/rings/number_field/maps.py --- a/sage/rings/number_field/maps.py Sat Oct 24 20:06:53 2009 -0700 +++ b/sage/rings/number_field/maps.py Mon Dec 21 23:27:29 2009 -0800 @@ -213,30 +213,83 @@ NumberFieldIsomorphism.__init__(self, Hom(K, V)) def _call_(self, alpha): + """ + TESTS:: + + sage: K. = NumberField(x^5+2) + sage: R. = K[] + sage: D. = K.extension(y + a + 1) + sage: D(a) + a + sage: V, from_V, to_V = D.relative_vector_space() + sage: to_V(a) + (a) + sage: to_V(a^3) + (a^3) + sage: to_V(x0) + (-a - 1) + + sage: K. = QuadraticField(-3) + sage: L. = K.extension(x-5) + sage: L(a) + a + sage: a*b + 5*a + sage: b + 5 + sage: V, from_V, to_V = L.relative_vector_space() + sage: to_V(a) + (a) + """ # An element of a relative number field is represented # internally by an absolute polynomial over QQ. alpha = self.__K(alpha) + + # Pari doesn't return a valid relative polynomial from an + # absolute one in the case of relative degree one, so we work + # around it. (Bug submitted to Pari, fixed in svn unstable as + # of 1/22/08 according to Karim Belabas. Trac #1891 is a + # reminder to remove this workaround once we update Pari in + # Sage.) + if self.__K.relative_degree() == 1: + # Let K/F be our relative extension, let z be the absolute + # polynomial for K (which *need not* be the absolute + # polynomial for F!), and let pol be the relative + # polynomial for K/F. Then self.__rnf[10] returns a triple + # z, a, k, where z is as above, a is a polynomial + # representing the absolute generator gamma of F as a + # polynomial in a root of z, and k is an integer + # satisfying + # theta = beta + k*gamma + # where theta is a root of z, and beta is a root of pol. + # + # Now f is a polynomial representing our element alpha as + # a polynomial in theta. However, we need a polynomial g + # representing alpha in terms of gamma, so we simply use + # the relation above to write theta as a polynomial in + # gamma, and then we can simply evaluate f at that + # polynomial, and this is our g. The code below does + # exactly this -- all the business with subst and + # self.__x, self.__y is there to deal with Pari's + # frustrating representation of relative number fields and + # "variable priority." + # + f = alpha._pari_().lift() + z, a, k = self.__rnf[10] + beta = -self.__rnf[0][0] / self.__rnf[0][1] + theta = beta + k*self.__y + f_in_base = f.subst(self.__x, theta).lift().subst(self.__y, self.__x) + return self.__V([self.__K.base_field()(f_in_base)]) + + # f is the absolute polynomial that defines this number field + # element f = alpha.polynomial('x') - # f is the absolute polynomial that defines this number field element - if self.__K.relative_degree() == 1: - # Pari doesn't return a valid relative polynomial from an - # absolute one in the case of relative degree one. (Bug - # submitted to Pari, fixed in svn unstable as of 1/22/08 - # according to Karim Belabas. Trac #1891 is a reminder to - # remove this workaround once we update Pari in Sage.) - # However, in this case, the polynomial we want for g is - # just the polynomial we have for f, but as a polynomial - # in the generator of the base field of this relative - # extension. That generator is just -self.__rnf[0][0], so - # we can plug it in and it works. - g = f(-1*self.__rnf[0][0]) - else: - g = self.__rnf.rnfeltabstorel(pari(f)) + g = self.__rnf.rnfeltabstorel(pari(f)) # Now g is a relative polynomial that defines this element. # This g is a polynomial in a pari variable x with - # coefficients polynomials in a variable y. - # These coefficients define the coordinates of the - # vector we are constructing. + # coefficients polynomials in a variable y. These + # coefficients define the coordinates of the vector we are + # constructing. # The list v below has the coefficients that are the # components of the vector we are constructing, but each is diff -r f872a92f5f9a -r faaaf3d2cbae sage/rings/number_field/number_field.py --- a/sage/rings/number_field/number_field.py Sat Oct 24 20:06:53 2009 -0700 +++ b/sage/rings/number_field/number_field.py Mon Dec 21 23:27:29 2009 -0800 @@ -1037,6 +1037,13 @@ Isomorphism given by variable name change map: From: Number Field in z with defining polynomial x^2 + 3 To: Number Field in a with defining polynomial x^2 + 3) + + sage: K. = QuadraticField(-3) + sage: R. = K[] + sage: D. = K.extension(y) + sage: D_abs. = D.absolute_field() + sage: D_abs.structure()[0](y0) + -a """ try: if self.__to_self is not None: diff -r f872a92f5f9a -r faaaf3d2cbae sage/rings/number_field/number_field_element.pyx --- a/sage/rings/number_field/number_field_element.pyx Sat Oct 24 20:06:53 2009 -0700 +++ b/sage/rings/number_field/number_field_element.pyx Mon Dec 21 23:27:29 2009 -0800 @@ -66,7 +66,7 @@ from sage.modules.free_module_element import vector -from sage.libs.all import pari_gen +from sage.libs.all import pari_gen, pari from sage.libs.pari.gen import PariError from sage.structure.element cimport Element, generic_power_c @@ -2629,16 +2629,9 @@ self.__pari = {} if var is None: var = self.number_field().variable_name() - f = self.polynomial()._pari_() - gp = self.number_field().polynomial() - if gp.variable_name() != 'x': - gp = gp.change_variable_name('x') - g = gp._pari_() - gv = str(gp.parent().gen()) - if var != 'x': - f = f.subst("x",var) - if var != gv: - g = g.subst(gv, var) + + f = self.polynomial()._pari_().subst('x', var) + g = self.number_field().pari_polynomial().subst('x', var) h = f.Mod(g) self.__pari[var] = h return h @@ -3028,11 +3021,9 @@ pass except TypeError: self.__pari = {} - g = self.parent().pari_polynomial() + g = self.parent().pari_polynomial(var) f = self.polynomial()._pari_() f = f.subst('x', var) - g = g.subst('x', var) - h = f.Mod(g) h = f.Mod(g) self.__pari[var] = h return h diff -r f872a92f5f9a -r faaaf3d2cbae sage/rings/number_field/number_field_rel.py --- a/sage/rings/number_field/number_field_rel.py Sat Oct 24 20:06:53 2009 -0700 +++ b/sage/rings/number_field/number_field_rel.py Mon Dec 21 23:27:29 2009 -0800 @@ -185,7 +185,7 @@ EXAMPLES:: sage: K. = NumberField(x^3 - 2) - sage: t = K['x'].gen() + sage: t = polygen(K) sage: L. = K.extension(t^2+t+a); L Number Field in b with defining polynomial x^2 + x + a over its base field """ @@ -219,7 +219,7 @@ A relative extension of a relative extension:: - sage: x = ZZ['x'].0 + sage: x = polygen(ZZ) sage: k. = NumberField([x^2 + 2, x^2 + 1]) sage: l. = k.extension(x^2 + 3) sage: l @@ -233,20 +233,21 @@ Test that irreducibility testing is working:: - sage: x = ZZ['x'].0 + sage: x = polygen(ZZ) sage: K. = NumberField([x^2 + 2, x^2 + 3]) sage: K. = NumberField(x^2 + 2) - sage: L. = K.extension(K['x'].0^3 + 3*a) + sage: x = polygen(K) + sage: L. = K.extension(x^3 + 3*a) - sage: (K['x'].0^3 + 2*a).factor() + sage: (x^3 + 2*a).factor() (x - a) * (x^2 + a*x - 2) - sage: L. = K.extension(K['x'].0^3 + 2*a) + sage: L. = K.extension(x^3 + 2*a) Traceback (most recent call last): ... ValueError: defining polynomial (x^3 + 2*a) must be irreducible - sage: (K['x'].0^2 + 2).factor() + sage: (x^2 + 2).factor() (x - a) * (x + a) - sage: L. = K.extension(K['x'].0^2 + 2) + sage: L. = K.extension(x^2 + 2) Traceback (most recent call last): ... ValueError: defining polynomial (x^2 + 2) must be irreducible @@ -278,7 +279,7 @@ # polynomial in y to satisfy PARI's ordering requirements. if base.is_relative(): - abs_base = base.absolute_field('a') + abs_base = base.absolute_field(name+'0') from_abs_base, to_abs_base = abs_base.structure() else: abs_base = base @@ -311,7 +312,8 @@ self._assign_names(tuple(names), normalize=False) NumberField_generic.__init__(self, self.absolute_polynomial(), name=None, - latex_name=latex_name, check=False, embedding=embedding) + latex_name=latex_name, check=False, + embedding=embedding) v[0] = self._gen_relative() v = [self(x) for x in v] @@ -642,9 +644,9 @@ EXAMPLES:: - sage: x = QQ['x'].0 + sage: x = polygen(QQ) sage: K. = NumberField(x^3 - 2) - sage: t = K['x'].gen() + sage: t = polygen(K) sage: K.extension(t^2+t+a, 'b')._latex_() '( \\Bold{Q}[a]/(a^{3} - 2) )[b]/(b^{2} + b + a)' """ @@ -728,8 +730,9 @@ MORE EXAMPLES:: - sage: K. = NumberField(ZZ['x'].0^5 + 2, 'a') - sage: L. = K.extension(ZZ['x'].0^2 + 3*a, 'b') + sage: x = polygen(ZZ) + sage: K. = NumberField(x^5 + 2, 'a') + sage: L. = K.extension(x^2 + 3*a, 'b') sage: u = QQ['u'].gen() sage: t = u.parent()['t'].gen() @@ -806,25 +809,46 @@ sage: k(m.0^4) 9 - sage: K. = NumberField(ZZ['x'].0^2 + 2, 'a') - sage: L. = K.extension(ZZ['x'].0 - a, 'b') + sage: x = polygen(ZZ) + sage: K. = NumberField(x^2 + 2, 'a') + sage: L. = K.extension(x - a, 'b') sage: L(a) a sage: L(b+a) 2*a - sage: K. = NumberField(ZZ['x'].0^5 + 2, 'a') - sage: L. = K.extension(ZZ['x'].0 - a, 'b') + sage: K. = NumberField(x^5 + 2, 'a') + sage: L. = K.extension(x - a, 'b') sage: L(a) a sage: L(a**3) a^3 sage: L(a**2+b) a^2 + a - sage: L. = K.extension(ZZ['x'].0 + a/2, 'b') + sage: L. = K.extension(x + a/2, 'b') sage: L(a) a + sage: L(a).polynomial() + -2*x + sage: L(a).minpoly() + x - a + sage: L(a).absolute_minpoly() + x^5 + 2 sage: L(b) -1/2*a + sage: L(b).polynomial() + x + sage: L(b).absolute_minpoly() + x^5 - 1/16 + sage: L(b).minpoly() + x + 1/2*a + + :: + + sage: K. = NumberField(x^5+2) + sage: R. = K[] + sage: L. = K.extension(y + a**2) + sage: L(a) + a """ if isinstance(x, (int, long, rational.Rational, integer.Integer, pari_gen, @@ -950,14 +974,16 @@ def _rnfeltreltoabs(self, element, check=False): r""" - Return PARI's ``rnfeletreltoabs()``, but without requiring ``rnfinit()``. + Return PARI's ``rnfeltreltoabs()``, but without requiring + ``rnfinit()``. TESTS:: - sage: x = ZZ['x'].0 + sage: x = polygen(ZZ) sage: K. = NumberField(x^2 + 2) - sage: L. = K.extension(K['x'].0^3 + 3*a) - sage: M. = L.extension(L['x'].0^2 + 5*b) + sage: x = polygen(K) + sage: L. = K.extension(x^3 + 3*a) + sage: M. = L.extension(x^2 + 5*b) sage: L._rnfeltreltoabs(b^2 + a, check=True) -2/9*x^3 - 2 @@ -965,10 +991,13 @@ 1/625*x^6 """ z, a, k = self._pari_rnfequation() - # a is an alpha in extension such that z(alpha) = base_polynomial(alpha) = 0 - # extension_polynomial(beta) = 0 - # absolute_polynomial(theta) = 0 - # theta = beta + k alpha + # If the relative extension is L/K, then z is the absolute + # polynomial for L, a is a polynomial giving the absolute + # generator alpha of K as a polynomial in a root of z, and k + # is a small integer such that + # theta = beta + k*alpha, + # where theta is a root of z and beta is a root of the + # relative polynomial for L/K. pol = element.polynomial('y') t2 = pol(a).lift() if check: @@ -1226,7 +1255,7 @@ sage: K.base_field() Number Field in b with defining polynomial x^3 + 3 over its base field sage: K.absolute_base_field()[0] - Number Field in a with defining polynomial x^9 + 3*x^6 + 165*x^3 + 1 + Number Field in a0 with defining polynomial x^9 + 3*x^6 + 165*x^3 + 1 sage: K.base_field().absolute_field('z') Number Field in z with defining polynomial x^9 + 3*x^6 + 165*x^3 + 1 """ @@ -1241,16 +1270,19 @@ TESTS:: sage: K. = NumberField(x^2 + 2) - sage: L. = K.extension(K['x'].0^5 + 2*a) + sage: x = polygen(K) + sage: L. = K.extension(x^5 + 2*a) sage: L._pari_rnfequation() [x^10 + 8, Mod(-1/2*x^5, x^10 + 8), 0] + sage: x = polygen(ZZ) sage: NumberField(x^10 + 8, 'a').is_isomorphic(L) True Initialization is lazy enough to allow arithmetic in massive fields:: sage: K. = NumberField(x^10 + 2000*x + 100001) - sage: L. = K.extension(K['x'].0^10 + 2*a) + sage: x = polygen(K) + sage: L. = K.extension(x^10 + 2*a) sage: L._pari_rnfequation() [x^100 - 1024000*x^10 + 102401024, Mod(-1/2*x^10, x^100 - 1024000*x^10 + 102401024), 0] sage: a + b @@ -1339,7 +1371,7 @@ EXAMPLES:: - sage: x = ZZ['x'].0 + sage: x = polygen(ZZ) sage: K. = NumberField([x^2 + 2, x^2 + 3]); K Number Field in a with defining polynomial x^2 + 2 over its base field sage: K.pari_absolute_base_polynomial() @@ -1378,7 +1410,7 @@ sage: m. = k.extension(k['w']([i,0,1])) sage: m Number Field in z with defining polynomial w^2 + i over its base field - sage: m.pari_relative_polynomial () + sage: m.pari_relative_polynomial() Mod(1, y^2 + 1)*x^2 + Mod(y, y^2 + 1) sage: l. = m.extension(m['t'].0^2 + z) @@ -1977,7 +2009,7 @@ EXAMPLES:: - sage: x = QQ['x'].0 + sage: x = polygen(QQ) sage: K. = NumberField(x^2 + 1) sage: R. = PolynomialRing(K) sage: L = K.extension(t^5-t+a, 'b') @@ -2001,9 +2033,10 @@ EXAMPLES:: - sage: x = QQ['x'].0 + sage: x = polygen(QQ) sage: K. = NumberField(x^2+6) - sage: L. = K.extension(K['x'].gen()^2 + 3) ## extend by x^2+3 + sage: x = polygen(K) + sage: L. = K.extension(x^2 + 3) ## extend by x^2+3 sage: L.is_free() False """ @@ -2020,11 +2053,10 @@ EXAMPLES:: - sage: x = QQ['x'].0 - sage: K = NumberField(x^3 - 2, 'a') - sage: R = K['x'] - sage: L = K.extension(R.gen()^2 - K.gen(), 'b') - sage: b = L.gen() + sage: x = polygen(ZZ) + sage: K. = NumberField(x^3 - 2) + sage: R. = K[] + sage: L. = K.extension(y^2 - a) sage: L.lift_to_base(b^4) a^2 sage: L.lift_to_base(b)