Hi, Does anybody understand number field embeddings http://doc.sagemath.org/html/en/reference/number_fields/sage/rings/number_field/number_field_morphisms.html
and whether the default I in SR (i.e., NumberField(x^2+1)) is equipped with any embeddings. The above reference manual shows how to make number fields with embeddings, but I couldn't see how to query a number field for its embeddings. Obviously, I could figure this out by reading the source, but I'm don't have time right now. I'm curious though, since I want to talk about this in my grad course on Wednesday, and it's relevant to the bug. I'm posting worksheets like this for my class: https://cloud.sagemath.com/projects/b85dd1b3-3f5f-4c2d-b503-242612561b9e/files/lectures/2016-01-15/2016-01-15-part0-QQbar.sagews On Sun, Jan 17, 2016 at 10:27 AM, Volker Braun <vbraun.n...@gmail.com> wrote: > SR is special in that it wraps various other rings: > > sage: wrapped_qqbar = SR(QQbar(I)) > sage: wrapped_qqbar.parent() > Symbolic Ring > sage: wrapped_qqbar.pyobject() > 1*I > sage: wrapped_qqbar.pyobject().parent() > Algebraic Field > > You don't gain anything from wrapping stuff in SR; ideally you can avoid it > in your code. > > I opened http://trac.sagemath.org/ticket/19904 with a better testcase for > the segfault > > sage: bool(SR(QQbar(I)) == I) > terminate called after throwing an instance of 'std::runtime_error' > what(): > ------------------------------------------------------------------------ > /home/vbraun/Code/sage/local/lib/python2.7/site-packages/sage/ext/interrupt/interrupt.so(+0x3da5)[0x7f06b9187da5] > /home/vbraun/Code/sage/local/lib/python2.7/site-packages/sage/ext/interrupt/interrupt.so(+0x3df7)[0x7f06b9187df7] > /home/vbraun/Code/sage/local/lib/python2.7/site-packages/sage/ext/interrupt/interrupt.so(+0x644c)[0x7f06b918a44c] > /lib64/libpthread.so.0(+0x109f0)[0x7f06c8b0e9f0] > /lib64/libc.so.6(gsignal+0x38)[0x7f06c8068a98] > /lib64/libc.so.6(abort+0x16a)[0x7f06c806a69a] > /home/vbraun/Code/sage/local/lib/libstdc++.so.6(__gnu_cxx::__verbose_terminate_handler()+0x15d)[0x7f06acab2ccd] > /home/vbraun/Code/sage/local/lib/libstdc++.so.6(+0x5dd46)[0x7f06acab0d46] > /home/vbraun/Code/sage/local/lib/libstdc++.so.6(+0x5dd91)[0x7f06acab0d91] > /home/vbraun/Code/sage/local/lib/libstdc++.so.6(+0x5dfa9)[0x7f06acab0fa9] > /home/vbraun/Code/sage/local/lib/libpynac.so.0(py_error(char > const*)+0x70)[0x7f069376ede0] > /home/vbraun/Code/sage/local/lib/libpynac.so.0(GiNaC::numeric::numeric(_object*, > bool)+0x1b8)[0x7f069376f9e8] > /home/vbraun/Code/sage/local/lib/libpynac.so.0(GiNaC::numeric::add(GiNaC::numeric > const&) const+0x1fa)[0x7f0693776c6a] > /home/vbraun/Code/sage/local/lib/libpynac.so.0(GiNaC::operator+(GiNaC::numeric > const&, GiNaC::numeric const&)+0x9)[0x7f069377a3a9] > /home/vbraun/Code/sage/local/lib/libpynac.so.0(GiNaC::numeric::add_dyn(GiNaC::numeric > const&) const+0x39)[0x7f0693770889] > /home/vbraun/Code/sage/local/lib/libpynac.so.0(GiNaC::expairseq::combine_overall_coeff(GiNaC::ex > const&)+0x1a)[0x7f06936d163a] > /home/vbraun/Code/sage/local/lib/libpynac.so.0(GiNaC::expairseq::construct_from_2_ex(GiNaC::ex > const&, GiNaC::ex const&)+0x375)[0x7f06936d6515] > /home/vbraun/Code/sage/local/lib/libpynac.so.0(GiNaC::add::add(GiNaC::ex > const&, GiNaC::ex const&)+0x63)[0x7f06936a5a73] > /home/vbraun/Code/sage/local/lib/libpynac.so.0(GiNaC::operator-(GiNaC::ex > const&, GiNaC::ex const&)+0x5f)[0x7f069377b0af] > /home/vbraun/Code/sage/local/lib/libpynac.so.0(GiNaC::relational::decide() > const+0x46)[0x7f069378b886] > /home/vbraun/Code/sage/local/lib/python2.7/site-packages/sage/symbolic/expression.so(+0x9ec48)[0x7f0693387c48] > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyObject_IsTrue+0x37)[0x7f06c8dbaa57] > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(+0x54e4e)[0x7f06c8d6fe4e] > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(+0xbcb63)[0x7f06c8dd7b63] > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyObject_Call+0x43)[0x7f06c8d6dc43] > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x3b2e)[0x7f06c8e2228e] > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x830)[0x7f06c8e255a0] > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalCode+0x19)[0x7f06c8e256c9] > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x58de)[0x7f06c8e2403e] > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x830)[0x7f06c8e255a0] > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5936)[0x7f06c8e24096] > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x830)[0x7f06c8e255a0] > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5936)[0x7f06c8e24096] > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x830)[0x7f06c8e255a0] > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5936)[0x7f06c8e24096] > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x830)[0x7f06c8e255a0] > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5936)[0x7f06c8e24096] > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x830)[0x7f06c8e255a0] > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5936)[0x7f06c8e24096] > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x830)[0x7f06c8e255a0] > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5936)[0x7f06c8e24096] > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x830)[0x7f06c8e255a0] > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalCode+0x19)[0x7f06c8e256c9] > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyRun_FileExFlags+0x8a)[0x7f06c8e4949a] > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyRun_SimpleFileExFlags+0xd7)[0x7f06c8e4aa27] > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(Py_Main+0xc25)[0x7f06c8e60c25] > /lib64/libc.so.6(__libc_start_main+0xf0)[0x7f06c8054580] > python(_start+0x29)[0x400769] > > > > > > On Sunday, January 17, 2016 at 5:39:43 PM UTC, Emmanuel Charpentier wrote: >> >> (Note : Question also asked on ask.sagemath.org, but crossposted because I >> found a way to CRASH sage...). >> I'm trying to understand coercions, and I'm hitting (repeatedly) something >> that I do not understand. >> >> Let's try to find thge roots of a polynom. We can try equation solving (of >> a quartic, no less) : >> >> sage: w = x^4 - (1+3*i)*x^3 - (2-4*i)*x^2 + (6-2*i)*x - 4 - 4*i >> sage: S1=[t.rhs() for t in solve(w,x)];S1 >> [-1/2*sqrt(2*I) + 3/2*I - 1/2, 1/2*sqrt(2*I) + 3/2*I - 1/2, -I + 1, I + 1] >> sage: bool(sqrt(2*I)==1+I) >> True >> >> Or we can try the roots of a polynomial : >> >> sage: S2=[SR(t[0]) for t in w.roots(ring=QQbar)];S2 >> [-1 + 1*I, 2*I, 1 - 1*I, 1 + 1*I] >> sage: bool(sqrt(2*I)==1+I) >> True >> sage: S1R=[t.subs({sqrt(2*I):1+I}) for t in S1];S1R >> [I - 1, 2*I, -I + 1, I + 1] >> >> So far, so good. I'm convinced that these two solution lists are one and >> the same. But I can't find a way to convince Sage: >> >> sage: map(lambda t,u:t-u, S1R, S2) >> >> --------------------------------------------------------------------------- >> TypeError Traceback (most recent call >> last) >> <ipython-input-6-be2e4183b522> in <module>() >> ----> 1 map(lambda t,u:t-u, S1R, S2) >> >> <ipython-input-6-be2e4183b522> in <lambda>(t, u) >> ----> 1 map(lambda t,u:t-u, S1R, S2) >> >> /usr/local/sage-7.0/src/sage/structure/element.pyx in >> sage.structure.element.RingElement.__sub__ >> (/usr/local/sage-7.0/src/build/cythonized/sage/structure/element.c:15995)() >> 1665 cdef long n >> 1666 if have_same_parent_c(left, right): >> -> 1667 return >> (<ModuleElement>left)._sub_(<ModuleElement>right) >> 1668 if PyInt_CheckExact(right): >> 1669 n = PyInt_AS_LONG(right) >> >> /usr/local/sage-7.0/src/sage/symbolic/expression.pyx in >> sage.symbolic.expression.Expression._sub_ >> (/usr/local/sage-7.0/src/build/cythonized/sage/symbolic/expression.cpp:20844)() >> 2950 relational_operator(_right._gobj)) >> 2951 else: >> -> 2952 x = gsub(left._gobj, _right._gobj) >> 2953 return new_Expression_from_GEx(left._parent, x) >> 2954 >> >> /usr/local/sage-7.0/src/sage/structure/element.pyx in >> sage.structure.element.RingElement.__add__ >> (/usr/local/sage-7.0/src/build/cythonized/sage/structure/element.c:15852)() >> 1649 elif PyInt_CheckExact(left): >> 1650 return >> (<RingElement>right)._add_long(PyInt_AS_LONG(left)) >> -> 1651 return coercion_model.bin_op(left, right, add) >> 1652 >> 1653 cdef RingElement _add_long(self, long n): >> >> /usr/local/sage-7.0/src/sage/structure/coerce.pyx in >> sage.structure.coerce.CoercionModel_cache_maps.bin_op >> (/usr/local/sage-7.0/src/build/cythonized/sage/structure/coerce.c:9736)() >> 1067 # We should really include the underlying error. >> 1068 # This causes so much headache. >> -> 1069 raise TypeError(arith_error_message(x,y,op)) >> 1070 >> 1071 cpdef canonical_coercion(self, x, y): >> >> TypeError: unsupported operand parent(s) for '+': 'Number Field in I with >> defining polynomial x^2 + 1' and 'Algebraic Field' >> >> I do not understand this error : both S1R and S2 are composed of things >> belonging to SR : >> >> sage: map(lambda t:type(t), S1R) >> [<type 'sage.symbolic.expression.Expression'>, >> <type 'sage.symbolic.expression.Expression'>, >> <type 'sage.symbolic.expression.Expression'>, >> <type 'sage.symbolic.expression.Expression'>] >> sage: map(lambda t:type(t), S2) >> [<type 'sage.symbolic.expression.Expression'>, >> <type 'sage.symbolic.expression.Expression'>, >> <type 'sage.symbolic.expression.Expression'>, >> <type 'sage.symbolic.expression.Expression'>] >> >> The reverse conversion works, however : >> >> sage: map(lambda t,u:t-u, [QQbar(t) for t in S1R], [QQbar(t) for t in S2]) >> [0, 0, 0, 0] >> >> Worse : testing directly for equality CRASHES Sage : >> >> sage: map(lambda t,u:bool(t==u), S1R, S2) >> terminate called after throwing an instance of 'std::runtime_error' >> what(): >> ------------------------------------------------------------------------ >> >> /usr/local/sage-7.0/local/lib/python2.7/site-packages/sage/ext/interrupt/interrupt.so(+0x3c35)[0x7f3d47047c35] >> >> /usr/local/sage-7.0/local/lib/python2.7/site-packages/sage/ext/interrupt/interrupt.so(+0x3c87)[0x7f3d47047c87] >> >> /usr/local/sage-7.0/local/lib/python2.7/site-packages/sage/ext/interrupt/interrupt.so(+0x608c)[0x7f3d4704a08c] >> /lib/x86_64-linux-gnu/libpthread.so.0(+0x10670)[0x7f3d4edc4670] >> /lib/x86_64-linux-gnu/libc.so.6(gsignal+0x37)[0x7f3d4e337657] >> /lib/x86_64-linux-gnu/libc.so.6(abort+0x16a)[0x7f3d4e338a2a] >> >> /usr/lib/x86_64-linux-gnu/libstdc++.so.6(_ZN9__gnu_cxx27__verbose_terminate_handlerEv+0x15d)[0x7f3d3a9ad35d] >> /usr/lib/x86_64-linux-gnu/libstdc++.so.6(+0x8d3b6)[0x7f3d3a9ab3b6] >> /usr/lib/x86_64-linux-gnu/libstdc++.so.6(+0x8d401)[0x7f3d3a9ab401] >> /usr/lib/x86_64-linux-gnu/libstdc++.so.6(+0x8d619)[0x7f3d3a9ab619] >> >> /usr/local/sage-7.0/local/lib/libpynac.so.2(_Z8py_errorPKc+0x49)[0x7f3d214029c9] >> >> /usr/local/sage-7.0/local/lib/libpynac.so.2(_ZN5GiNaC7numericC1EP7_objectb+0x1b8)[0x7f3d21403508] >> >> /usr/local/sage-7.0/local/lib/libpynac.so.2(_ZNK5GiNaC7numeric3addERKS0_+0x1c2)[0x7f3d214094f2] >> >> /usr/local/sage-7.0/local/lib/libpynac.so.2(_ZN5GiNaCplERKNS_7numericES2_+0x9)[0x7f3d2140c569] >> >> /usr/local/sage-7.0/local/lib/libpynac.so.2(_ZNK5GiNaC7numeric7add_dynERKS0_+0x2d)[0x7f3d2140405d] >> >> /usr/local/sage-7.0/local/lib/libpynac.so.2(_ZN5GiNaC9expairseq21combine_overall_coeffERKNS_2exE+0x15)[0x7f3d2136f945] >> >> /usr/local/sage-7.0/local/lib/libpynac.so.2(_ZN5GiNaC9expairseq19construct_from_2_exERKNS_2exES3_+0x27d)[0x7f3d21374bed] >> >> /usr/local/sage-7.0/local/lib/libpynac.so.2(_ZN5GiNaC3addC1ERKNS_2exES3_+0x63)[0x7f3d21346633] >> >> /usr/local/sage-7.0/local/lib/libpynac.so.2(_ZN5GiNaCmiERKNS_2exES2_+0x5f)[0x7f3d2140d14f] >> >> /usr/local/sage-7.0/local/lib/libpynac.so.2(_ZNK5GiNaC10relational6decideEv+0x3e)[0x7f3d2141bede] >> >> /usr/local/sage-7.0/local/lib/python2.7/site-packages/sage/symbolic/expression.so(+0xa7108)[0x7f3d21039108] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyObject_IsTrue+0x37)[0x7f3d4f06fe17] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(+0x54eae)[0x7f3d4f025eae] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(+0xbc623)[0x7f3d4f08d623] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyObject_Call+0x43)[0x7f3d4f023a73] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x3a6e)[0x7f3d4f0d634e] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x81c)[0x7f3d4f0d928c] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(+0x835ac)[0x7f3d4f0545ac] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyObject_Call+0x43)[0x7f3d4f023a73] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_CallObjectWithKeywords+0x47)[0x7f3d4f0d22e7] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(+0xfd564)[0x7f3d4f0ce564] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5c8e)[0x7f3d4f0d856e] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x81c)[0x7f3d4f0d928c] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalCode+0x19)[0x7f3d4f0d93a9] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5234)[0x7f3d4f0d7b14] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x81c)[0x7f3d4f0d928c] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5a42)[0x7f3d4f0d8322] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x81c)[0x7f3d4f0d928c] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5a42)[0x7f3d4f0d8322] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x81c)[0x7f3d4f0d928c] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5a42)[0x7f3d4f0d8322] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x81c)[0x7f3d4f0d928c] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5a42)[0x7f3d4f0d8322] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x81c)[0x7f3d4f0d928c] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5a42)[0x7f3d4f0d8322] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x81c)[0x7f3d4f0d928c] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5a42)[0x7f3d4f0d8322] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x81c)[0x7f3d4f0d928c] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalCode+0x19)[0x7f3d4f0d93a9] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyRun_FileExFlags+0x8a)[0x7f3d4f0fca9a] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyRun_SimpleFileExFlags+0xd7)[0x7f3d4f0fde47] >> >> /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(Py_Main+0xc3e)[0x7f3d4f1143ee] >> /lib/x86_64-linux-gnu/libc.so.6(__libc_start_main+0xf0)[0x7f3d4e324870] >> python(_start+0x29)[0x4006f9] >> ------------------------------------------------------------------------ >> Attaching gdb to process id 11241. >> >> Saved trace to /home/charpent/.sage/crash_logs/sage_crash_qQFZhv.log >> ------------------------------------------------------------------------ >> Unhandled SIGABRT: An abort() occurred in Sage. >> This probably occurred because a *compiled* component of Sage has a bug >> in it and is not properly wrapped with sig_on(), sig_off(). >> Sage will now terminate. >> ------------------------------------------------------------------------ >> Abandon >> >> > -- > You received this message 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