+ i]Rt^RIHt^RIHt^RIHt^RIHt^RI H t ^RI H t ^RI Ht^RIHt^R IHtHt^R IHt^R IHtHt^R IHtHtR t!RR]4tRt]R4tRRlt Rt!Rt"Rt#Rt$]RRl4t%R#)zPrime ideals in number fields. )Poly)FF)QQ)ZZ) DomainMatrix)CoercionFailed)IntegerPowerable)public) round_twonilradical_mod_p)StructureError)ModuleEndomorphism find_min_poly) coeff_searchsupplement_a_subspacecRpRpVP4'gRpM1VP4'gRpMVP4'gRpVe\W,4hR#)a Several functions in this module accept an argument which is to be a :py:class:`~.Submodule` representing the maximal order in a number field, such as returned by the :py:func:`~sympy.polys.numberfields.basis.round_two` algorithm. We do not attempt to check that the given ``Submodule`` actually represents a maximal order, but we do check a basic set of formal conditions that the ``Submodule`` must satisfy, at a minimum. The purpose is to catch an obviously ill-formed argument. z4The submodule representing the maximal order should Nz'be a direct submodule of a power basis.zhave 1 as its first generator.z>> from sympy import cyclotomic_poly, QQ >>> from sympy.abc import x, zeta >>> T = cyclotomic_poly(7, x) >>> K = QQ.algebraic_field((T, zeta)) >>> P = K.primes_above(11) >>> print(P[0].repr()) [ (11, x**3 + 5*x**2 + 4*x - 1) e=1, f=3 ] >>> print(P[0].repr(field_gen=zeta)) [ (11, zeta**3 + 5*zeta**2 + 4*zeta - 1) e=1, f=3 ] >>> print(P[0].repr(field_gen=zeta, just_gens=True)) (11, zeta**3 + 5*zeta**2 + 4*zeta - 1) Parameters ========== field_gen : :py:class:`~.Symbol`, ``None``, optional (default=None) The symbol to use for the generator of the field. This will appear in our representation of ``self.alpha``. If ``None``, we use the variable of the defining polynomial of ``self.ZK``. just_gens : bool, optional (default=False) If ``True``, just print the "(p, alpha)" part, showing "just the generators" of the prime ideal. Otherwise, print a string of the form "[ (p, alpha) e=..., f=... ]", giving the ramification index and inertia degree, along with the generators. )xr)z)/r+r*z[ z e=z, f=z ]) rparentTgenrr r$r!str numeratorr-denom) r% field_gen just_gensrr r$r! alpha_repgenss &&& rreprPrimeIdeal.reprUsB55!1!1!5!5 TVVTVV;!)4<<>? ;;?I;b 6I1#R {!$ KD6QCtA3b))rc"VP4#N)r@r.s&r__repr__PrimeIdeal.__repr__syy{rc VPVP,VPVP,,pRVnRVnV#)a Represent this prime ideal as a :py:class:`~.Submodule`. Explanation =========== The :py:class:`~.PrimeIdeal` class serves to bundle information about a prime ideal, such as its inertia degree, ramification index, and two-generator representation, as well as to offer helpful methods like :py:meth:`~.PrimeIdeal.valuation` and :py:meth:`~.PrimeIdeal.test_factor`. However, in order to be added and multiplied by other ideals or rational numbers, it must first be converted into a :py:class:`~.Submodule`, which is a class that supports these operations. In many cases, the user need not perform this conversion deliberately, since it is automatically performed by the arithmetic operator methods :py:meth:`~.PrimeIdeal.__add__` and :py:meth:`~.PrimeIdeal.__mul__`. Raising a :py:class:`~.PrimeIdeal` to a non-negative integer power is also supported. Examples ======== >>> from sympy import Poly, cyclotomic_poly, prime_decomp >>> T = Poly(cyclotomic_poly(7)) >>> P0 = prime_decomp(7, T)[0] >>> print(P0**6 == 7*P0.ZK) True Note that, on both sides of the equation above, we had a :py:class:`~.Submodule`. In the next equation we recall that adding ideals yields their GCD. This time, we need a deliberate conversion to :py:class:`~.Submodule` on the right: >>> print(P0 + 7*P0.ZK == P0.as_submodule()) True Returns ======= :py:class:`~.Submodule` Will be equal to ``self.p * self.ZK + self.alpha * self.ZK``. See Also ======== __add__ __mul__ FT)rrr _starts_with_unity_is_sq_maxrank_HNF)r%Ms& r as_submodulePrimeIdeal.as_submodules>n FFTWW tzzDGG3 3$#rc|\V\4'd"VP4VP48H#\#rC) isinstancerrJNotImplementedr%others&&r__eq__PrimeIdeal.__eq__s2 eZ ( ($$&%*<*<*>> >rc 0VP4V,#)zt Convert to a :py:class:`~.Submodule` and add to another :py:class:`~.Submodule`. See Also ======== as_submodule rJrOs&&r__add__PrimeIdeal.__add__  "U**rc 0VP4V,#)z Convert to a :py:class:`~.Submodule` and multiply by another :py:class:`~.Submodule` or a rational number. See Also ======== as_submodule rTrOs&&r__mul__PrimeIdeal.__mul__rWrcVP#rC)rr.s&r _zeroth_powerPrimeIdeal._zeroth_powers wwrcV#rCr.s&r _first_powerPrimeIdeal._first_powers rc VPf2\VPVP.VP4VnVP#)a Compute a test factor for this prime ideal. Explanation =========== Write $\mathfrak{p}$ for this prime ideal, $p$ for the rational prime it divides. Then, for computing $\mathfrak{p}$-adic valuations it is useful to have a number $\beta \in \mathbb{Z}_K$ such that $p/\mathfrak{p} = p \mathbb{Z}_K + \beta \mathbb{Z}_K$. Essentially, this is the same as the number $\Psi$ (or the "reagent") from Kummer's 1847 paper (*Ueber die Zerlegung...*, Crelle vol. 35) in which ideal divisors were invented. )r"_compute_test_factorrr rr.s&r test_factorPrimeIdeal.test_factors;    $ 4TVVdjj\477 SD    rc \W4#)z Compute the $\mathfrak{p}$-adic valuation of integral ideal I at this prime ideal. Parameters ========== I : :py:class:`~.Submodule` See Also ======== prime_valuation )prime_valuation)r%Is&&rr#PrimeIdeal.valuations q''rc @VP4PV4#)aP Reduce a :py:class:`~.PowerBasisElement` to a "small representative" modulo this prime ideal. Parameters ========== elt : :py:class:`~.PowerBasisElement` The element to be reduced. Returns ======= :py:class:`~.PowerBasisElement` The reduced element. See Also ======== reduce_ANP reduce_alg_num .Submodule.reduce_element )rJreduce_element)r%elts&&rrkPrimeIdeal.reduce_elements2  "11#66rc VPPPV4pVPV4pVP 4#)a+ Reduce an :py:class:`~.ANP` to a "small representative" modulo this prime ideal. Parameters ========== elt : :py:class:`~.ANP` The element to be reduced. Returns ======= :py:class:`~.ANP` The reduced element. See Also ======== reduce_element reduce_alg_num .Submodule.reduce_element )rr6element_from_ANPrkto_ANPr%arlreds&& r reduce_ANPPrimeIdeal.reduce_ANP*s82ggnn--a0!!#&zz|rc VPPPV4pVPV4pVP \ \ VPP4444#)aK Reduce an :py:class:`~.AlgebraicNumber` to a "small representative" modulo this prime ideal. Parameters ========== elt : :py:class:`~.AlgebraicNumber` The element to be reduced. Returns ======= :py:class:`~.AlgebraicNumber` The reduced element. See Also ======== reduce_element reduce_ANP .Submodule.reduce_element ) rr6element_from_alg_numrk field_elementlistreversedQQ_colflatrqs&& rreduce_alg_numPrimeIdeal.reduce_alg_numGsP2ggnn11!4!!#&tHSZZ__->$?@AAr)rr"r r$r!rrC)NF)__name__ __module__ __qualname____firstlineno____doc__r&r/propertyr,r@rDrJrQrU__radd__rY__rmul__r\r`rdr#rkrtr}__static_attributes____classdictcell__) __classdict__s@rrr)s@45 ##)*V;z +H +H!(($76:BBrrc\V4VP4pVUu.uF#qCPV4PVR7NK% pp\P !^VP 3\V44P!V!pVP4R,P4pVPVPVP\4,VPR7pV#uupi)a Compute the test factor for a :py:class:`~.PrimeIdeal` $\mathfrak{p}$. Parameters ========== p : int The rational prime $\mathfrak{p}$ divides gens : list of :py:class:`PowerBasisElement` A complete set of generators for $\mathfrak{p}$ over *ZK*, EXCEPT that an element equivalent to rational *p* can and should be omitted (since it has no effect except to waste time). ZK : :py:class:`~.Submodule` The maximal order where the prime ideal $\mathfrak{p}$ lives. Returns ======= :py:class:`~.PowerBasisElement` References ========== .. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.* (See Proposition 4.8.15.) modulusr;)NNN)rendomorphism_ringinner_endomorphismmatrixrzerosr2rvstack nullspace transposer6 convert_torr;) rr?rEgmatricesBr5betas &&& rrcrces</r2 ACGH4a$$Q'..q.94HHArtt9be,33X>A d%%'A 99RYYb!119 BD KIs)C-cVPVPr2VPVPVPrepVP \ 4P4VP,V,VP, pVP \4pVP4pW,^8wd^#VP4p Wd,VP4,p W,^8Hp ^p WW,p\V4Fdp VPVRV 3,VR7pW,pVPV4P4p\V4FpW,WV 3&K Kf Wt^, V^, 3,PV,^8wdV #Wr, pV 'dVP \4pMVP \4pV ^, p K \ dT #i;i)a Compute the *P*-adic valuation for an integral ideal *I*. Examples ======== >>> from sympy import QQ >>> from sympy.polys.numberfields import prime_valuation >>> K = QQ.cyclotomic_field(5) >>> P = K.primes_above(5) >>> ZK = K.maximal_order() >>> print(prime_valuation(25*ZK, P[0])) 8 Parameters ========== I : :py:class:`~.Submodule` An integral ideal whose valuation is desired. P : :py:class:`~.PrimeIdeal` The prime at which to compute the valuation. Returns ======= int See Also ======== .PrimeIdeal.valuation References ========== .. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.* (See Algorithm 4.8.17.) rr)rrr2rr;rrinvrdetrdranger6 representr|elementr)rhPrrr2WdADrr!need_complete_testvjcis&& rrgrgsT CCrddBIIrxx!A R)A-7A RA Auz ==?D !%%'A%1* A  EqA !AqD' +A IA Q$$&A1X$Q$  UAE\? " "Q &! +  H E  LL$  R A Q "  H  s3G)) G87G8Nca\V4VPpVPp\;QJdV3RlV4F 'dK RM RM!V3RlV44'dVP 4#VfAVe SV,pM2\ VP V4PP44pVP4pVR,Uu.uF pSV,NK p pW, p \\V 4^4p V FOp \R\W444p V PV4V,p V S,^8wgKFV S,u# R#uupi)a Given a set of *ZK*-generators of a prime ideal, compute a set of just two *ZK*-generators for the same ideal, one of which is *p* itself. Parameters ========== gens : list of :py:class:`PowerBasisElement` Generators for the prime ideal over *ZK*, the ring of integers of the field $K$. ZK : :py:class:`~.Submodule` The maximal order in $K$. p : int The rational prime divided by the prime ideal. f : int, optional The inertia degree of the prime ideal, if known. Np : int, optional The norm $p^f$ of the prime ideal, if known. NOTE: There is no reason to supply both *f* and *Np*. Either one will save us from having to compute the norm *Np* ourselves. If both are known, *Np* is preferred since it saves one exponentiation. Returns ======= :py:class:`~.PowerBasisElement` representing a single algebraic integer alpha such that the prime ideal is equal to ``p*ZK + alpha*ZK``. References ========== .. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.* (See Algorithm 4.7.10.) c3R<"TFqS,P^4xK R#5i)rN)equiv).0rrs& r _two_elt_rep..s *TE==  T$'FTN:NNc36"TFwrW,xK R#5irCr_)rcibetais& rrr's;lBHHls)rr6r7allzeroabssubmodule_from_gensrrbasis_element_pullbacksrlensumzipnorm)r?rrr!Nppbr7omegaomrsearchrr r2s&&f&& r _two_elt_reprsP/r2 B A s *T *sss *T ***wwy z =ABR++D188<<>?B  & & (E 9 %9RAbDD9D %LD #d)Q 'F ;c!l;; JJqMR  q5A:19  &sEcVPPp\W R7pVP4wrE\ V4^8XdGV^,^,^8Xd2\ WVPP 4VP^4.#VUUu.uFIwrg\ WVPP\V\R74VP4V4NKK upp#uuppi)a' Compute the decomposition of rational prime *p* in the ring of integers *ZK* (given as a :py:class:`~.Submodule`), in the "easy case", i.e. the case where *p* does not divide the index of $\theta$ in *ZK*, where $\theta$ is the generator of the ``PowerBasis`` of which *ZK* is a ``Submodule``. rdomain) r6r7r factor_listrrrr2element_from_polyrdegree)rrr7T_barlcfltr$s&& r_prime_decomp_easy_caser1s A  E    FB 2w!|1aA 2")).."2BDD!<==  ryy224"3EFxxz1 &  sAC#cFaVPpVPwrEV^8XdVPV\4pM,VP VPV\4R,4pVP^,V8d7\ VP \S444P \4pVPV4pVP4VPV4p\VV3Rl4pVPSR7p V P4'gQhW3#)a Parameters ========== I : :py:class:`~.Module` An ideal of ``ZK/pZK``. p : int The rational prime being factored. ZK : :py:class:`~.Submodule` The maximal order. Returns ======= Pair ``(N, G)``, where: ``N`` is a :py:class:`~.Module` representing the kernel of the map ``a |--> a**p - a`` on ``(O/pO)/I``, guaranteed to be a module with unity. ``G`` is a :py:class:`~.Module` representing a basis for the separable algebra ``A = O/I`` (see Cohen). c$<VS,V, #rCr_)r5rs&r._prime_decomp_compute_kernel..ss !Q$(rr)rr)rshapeeyerhstackrrrsubmodule_from_matrixcompute_mult_tabdiscard_beforer kernelr) rhrrrr2rrGphiNs &f& r_prime_decomp_compute_kernelrDs2 A 77DA  Av EE!RL HHQUU1b\$' (wwqzA~ !!,,r!u"5 6 A A" E   #A  A Q 2 3C 1 A    4KrcTVPPwr4W4, pVPVP,p\VP^,4Uu.uF(qrPVRV3,VPR7NK* pp\ WWR7p \ W!W4#uupi)a} We have reached the case where we have a maximal (hence prime) ideal *I*, which we know because the quotient ``O/I`` is a field. Parameters ========== I : :py:class:`~.Module` An ideal of ``O/pO``. p : int The rational prime being factored. ZK : :py:class:`~.Submodule` The maximal order. Returns ======= :py:class:`~.PrimeIdeal` instance representing this prime rr)r!)rrrr6r;rr) rhrrmr2r!rrr?r s &&& r_prime_decomp_maximal_idealrys* 88>>DA A AHHA8=aggaj8I J8I1IIa1gRXXI .8ID J 1 *E bU && Ks.B%c JaaVPV8Xd!VPVJdVPVJgQhV!^4P4pVPVJgQh.o\V\ V4SR7pVP 4wrxV^,^,p VP V 4p V PV 4wrp V ^8XgQh\\\W,\R7PP444o\VV3Rl\\!S4444p^V, pW.p.pVFpVP4pVPVJgQhVP"P%\ V44P&!VP)4Uu.uF%pVV,P+\ V4R7NK' up!pVP-4P%\4pVP/V4pVP1V4K V#uupi)z Perform the step in the prime decomposition algorithm where we have determined the quotient ``ZK/I`` is _not_ a field, and we want to perform a non-trivial factorization of *I* by locating an idempotent element of ``ZK/I``. )powersrc3R<"TFpSV,SV,,xK R#5irCr_)rrr alpha_powerss& rr,_prime_decomp_split_ideal..s!;]qtLO##]r)r6 to_parentmodulerrrquogcdexryrzrrrepto_listrrrrrrbasis_elementscolumn columnspacerappend)rhrrrrr rrrm1m2UVreps1eps2idempsfactorsepsr$rrrHrrs&&&&& @@r_prime_decomp_split_idealrs 88r>ahh"nQ> > aDNN E <<1  LeRU<8A]]_FB AqB rBhhrlGA! 6M6 Xd16"-1199; <=A ;U3q6]; ;D t8D\FG MMOxx2~~ HH  1 & - -464E4E4G0 4GbQVOO2a5O )4G0   MMO & &r *  $ $Q 'q N 0 s%+H cfVfVf \R4hVe \V4VfVPPp/pVeVf\ WR7wr#VP 4pWc,pWp,^8wd \ W4#T;'g%VPV4;'g \W 4pV.p.p V'drVP4p \WV4wrV P^8Xd \WV4p V PV 4KV\WWV4wrVPW.4KyV #)a Compute the decomposition of rational prime *p* in a number field. Explanation =========== Ordinarily this should be accessed through the :py:meth:`~.AlgebraicField.primes_above` method of an :py:class:`~.AlgebraicField`. Examples ======== >>> from sympy import Poly, QQ >>> from sympy.abc import x, theta >>> T = Poly(x ** 3 + x ** 2 - 2 * x + 8) >>> K = QQ.algebraic_field((T, theta)) >>> print(K.primes_above(2)) [[ (2, x**2 + 1) e=1, f=1 ], [ (2, (x**2 + 3*x + 2)/2) e=1, f=1 ], [ (2, (3*x**2 + 3*x)/2) e=1, f=1 ]] Parameters ========== p : int The rational prime whose decomposition is desired. T : :py:class:`~.Poly`, optional Monic irreducible polynomial defining the number field $K$ in which to factor. NOTE: at least one of *T* or *ZK* must be provided. ZK : :py:class:`~.Submodule`, optional The maximal order for $K$, if already known. NOTE: at least one of *T* or *ZK* must be provided. dK : int, optional The discriminant of the field $K$, if already known. radical : :py:class:`~.Submodule`, optional The nilradical mod *p* in the integers of $K$, if already known. Returns ======= List of :py:class:`~.PrimeIdeal` instances. References ========== .. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.* (See Algorithm 6.2.9.) z)At least one of T or ZK must be provided.)radicals) ValueErrorrr6r7r discriminantrgetr poprr2rrrextend)rr7rdKradicalrdT f_squaredstackprimesrhrrrI1I2s&&&&& r prime_decomprs n yRZDEE ~226y IIKKH zRZ10  BI}&q--CCaCC,FB LL" " Mr)NN)NNNN)&rsympy.polys.polytoolsrsympy.polys.domains.finitefieldr!sympy.polys.domains.rationalfieldrsympy.polys.domains.integerringr!sympy.polys.matrices.domainmatrixrsympy.polys.polyerrorsrsympy.polys.polyutilsrsympy.utilities.decoratorr basisr r exceptionsr modulesr r utilitiesrrrrrcrgrrrrrrr_rrrs%&.0.:12,.&6:,0yB!yBx +\U U pBJ&2j':'TOOr