*h] SrSSKJr SSKJr SSKJr SSKJr SSK J r SSK J r SSK Jr SS KJr S S KJrJr S S KJr S S KJrJr S SKJrJr Sr"SS\5rSr\S5rSSjr Sr!Sr"Sr#Sr$\SSj5r%g)zPrime ideals in number fields. )Poly)FF)QQ)ZZ) DomainMatrix)CoercionFailed)IntegerPowerable)public) round_twonilradical_mod_p)StructureError)ModuleEndomorphism find_min_poly) coeff_searchsupplement_a_subspacecSnSnUR5(dSnO/UR5(dSnOUR5(dSnUb [X-5eg)a Several functions in this module accept an argument which is to be a :py:class:`~.Submodule` representing the maximal order in a number field, such as returned by the :py:func:`~sympy.polys.numberfields.basis.round_two` algorithm. We do not attempt to check that the given ``Submodule`` actually represents a maximal order, but we do check a basic set of formal conditions that the ``Submodule`` must satisfy, at a minimum. The purpose is to catch an obviously ill-formed argument. z4The submodule representing the maximal order should Nz'be a direct submodule of a power basis.zhave 1 as its first generator.z>> from sympy import cyclotomic_poly, QQ >>> from sympy.abc import x, zeta >>> T = cyclotomic_poly(7, x) >>> K = QQ.algebraic_field((T, zeta)) >>> P = K.primes_above(11) >>> print(P[0].repr()) [ (11, x**3 + 5*x**2 + 4*x - 1) e=1, f=3 ] >>> print(P[0].repr(field_gen=zeta)) [ (11, zeta**3 + 5*zeta**2 + 4*zeta - 1) e=1, f=3 ] >>> print(P[0].repr(field_gen=zeta, just_gens=True)) (11, zeta**3 + 5*zeta**2 + 4*zeta - 1) Parameters ========== field_gen : :py:class:`~.Symbol`, ``None``, optional (default=None) The symbol to use for the generator of the field. This will appear in our representation of ``self.alpha``. If ``None``, we use the variable of the defining polynomial of ``self.ZK``. just_gens : bool, optional (default=False) If ``True``, just print the "(p, alpha)" part, showing "just the generators" of the prime ideal. Otherwise, print a string of the form "[ (p, alpha) e=..., f=... ]", giving the ramification index and inertia degree, along with the generators. )xr r,z)/r.r-z[ z e=z, f=z ]) r!parentTgenr"r#r'r$str numeratorr0denom) r( field_gen just_gensr"r#r'r$ alpha_repgenss rreprPrimeIdeal.reprUsB5!1!1!5!5 TVVTVV;!)4<<>? ;;?I;b 6I1#R {!$ KD6QCtA3b))rc"UR5$N)rCr1s r__repr__PrimeIdeal.__repr__syy{rcURUR-URUR--nSUlSUlU$)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)r"r!r#_starts_with_unity_is_sq_maxrank_HNF)r(Ms r as_submodulePrimeIdeal.as_submodules>n FFTWW tzzDGG3 3$#rcz[U[5(a!UR5UR5:H$[$rF) isinstancerrMNotImplementedr(others r__eq__PrimeIdeal.__eq__s2 eZ ( ($$&%*<*<*>> >rc(UR5U-$)zt Convert to a :py:class:`~.Submodule` and add to another :py:class:`~.Submodule`. See Also ======== as_submodule rMrRs r__add__PrimeIdeal.__add__  "U**rc(UR5U-$)z Convert to a :py:class:`~.Submodule` and multiply by another :py:class:`~.Submodule` or a rational number. See Also ======== as_submodule rWrRs r__mul__PrimeIdeal.__mul__rZrcUR$rF)r!r1s r _zeroth_powerPrimeIdeal._zeroth_powers wwrcU$rFr1s r _first_powerPrimeIdeal._first_powers rcURc1[URUR/UR5UlUR$)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_factorr"r#r!r1s r test_factorPrimeIdeal.test_factors;    $ 4TVVdjj\477 SD    rc[X5$)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@UR5RU5$)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 )rMreduce_element)r(elts rrnPrimeIdeal.reduce_elements2  "11#66rcURRRU5nURU5nUR 5$)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 )r!r9element_from_ANPrnto_ANPr(aroreds r reduce_ANPPrimeIdeal.reduce_ANP*s82ggnn--a0!!#&zz|rcURRRU5nURU5nUR [ [ URR5555$)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 ) r!r9element_from_alg_numrn field_elementlistreversedQQ_colflatrts rreduce_alg_numPrimeIdeal.reduce_alg_numGsP2ggnn11!4!!#&tHSZZ__->$?@AAr)r!r%r#r'r$r"rF)NF)__name__ __module__ __qualname____firstlineno____doc__r)r2propertyr/rCrGrMrTrX__radd__r\__rmul__r_rcrgr&rnrwr__static_attributes__rbrrrr)sx@45 ##)*V;z +H +H!(($76:Brrc[U5 UR5nUVs/sH oCRU5RUS9PM" nn[R "SUR 4[U55R"U6nUR5SSS24R5nURURUR[5-URS9nU$s snf)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.) modulusrNr>)rendomorphism_ringinner_endomorphismmatrixrzerosr5rvstack nullspace transposer9 convert_torr>) r"rBr!EgmatricesBr8betas rrfrfes</r2 ACGH4a$$Q'..q.94HHArtt9be,33X>A ad%%'A 99RYYb!119 BD KIs'C#cbURURp2URURURpenUR [ 5R5UR-U-UR- nUR [5nUR5nX-S:wagUR5n Xd-UR5-n X-S:Hn Sn XW-n[U5HWn URUSS2U 4US9nX-nURU5R5n[U5H nXXU 4'M MY XtS- US- 4RU-S:waU $Xr- nU (aUR [5nOUR [5nU S- n M![ a U $f=f)a Compute the *P*-adic valuation for an integral ideal *I*. Examples ======== >>> from sympy import QQ >>> from sympy.polys.numberfields import prime_valuation >>> K = QQ.cyclotomic_field(5) >>> P = K.primes_above(5) >>> ZK = K.maximal_order() >>> print(prime_valuation(25*ZK, P[0])) 8 Parameters ========== I : :py:class:`~.Submodule` An integral ideal whose valuation is desired. P : :py:class:`~.PrimeIdeal` The prime at which to compute the valuation. Returns ======= int See Also ======== .PrimeIdeal.valuation References ========== .. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.* (See Algorithm 4.8.17.) rNrr )r"r!r5rr>rrinvrdetrgranger9 representrelementr)rkPr"r!r5WdADrr$need_complete_testvjcis rrjrjsT 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 Q7 ,"  H  s.F F.-F.Nc8^[U5 URnURn[U4SjU55(aUR 5$Uc;UbTU-nO2[ UR U5RR55nUR5nUSSVs/sHnTU-PM n nX- n [[U 5S5n U HBn [S[X555n U RU5U-n U T-S:wdM=U T-s $ gs snf)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.) c3J># UHoT-RS5v M g7f)rN)equiv).0rr"s r _two_elt_rep..s *TE==  Ts #Nr c3.# UH upX-v M g7frFrb)rcibetais rrr's;lBHlsr)rr9r:allzeroabssubmodule_from_gensrrbasis_element_pullbacksrlensumzipnorm)rBr!r"r$Nppbr:omegaomrsearchrr#r5s ` r _two_elt_reprsP/r2 B A *T ***wwy z =ABR++D188<<>?B  & & (E 9 %9RAbD9D %LD #d)Q 'F ;c!l;; JJqMR  q5A:19  &s$DcURRn[X S9nUR5upE[ U5S:Xa=USSS:Xa1[ XURR 5URS5/$UVVs/sHFupg[ XURR[U[S95UR5U5PMH snn$s snnf)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``. rr rdomain) r9r:r factor_listrrrr5element_from_polyrdegree)r"r!r:T_barlcfltr's r_prime_decomp_easy_caser1s A  E    FB 2w!|1aA 2")).."2BDD!<==  ryy224"3EFxxz1 &  sA Cc2^URnURupEUS:XaURU[5nO,UR URU[5SS2S45nURSU:a6[ UR [T555R [5nURU5nUR5 URU5n[UU4Sj5nURTS9n U R5(deX4$)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). rNr c>UT-U- $rFrb)r8r"s r._prime_decomp_compute_kernel..ss !Q$(rr)rshapeeyerhstackrrrsubmodule_from_matrixcompute_mult_tabdiscard_beforerkernelr) rkr"r!rr5rrGphiNs ` r_prime_decomp_compute_kernelrDs2 A 77DA  Av EE!RL HHQUU1b\!Q$' (wwqzA~ !!,,r!u"5 6 A A" E   #A  A Q 2 3C 1 A    4Krc.URRup4X4- nURUR-n[URS5Vs/sH#orRUSS2U4URS9PM% nn[ XXS9n [ X!X5$s snf)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 r Nr)r$)rrrr9r>rr) rkr"r!mr5r$rrrBr#s r_prime_decomp_maximal_idealrys* 88>>DA A AHHA8=aggaj8I J8I1IIa1gRXXI .8ID J 1 *E bU && Ks*Bc ^^URU:XaURULaURULdeU"S5R5nURULde/m[U[ U5TS9nUR 5upxUSSn UR U 5n U RU 5upn U S:Xde[[[X-[S9RR555m[UU4Sj[[!T5555nSU- nX/n/nUHnUR5nURULdeUR"R%[ U55R&"UR)5Vs/sHnUU-R+[ U5S9PM sn6nUR-5R%[5nUR/U5nUR1U5 M U$s snf)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``. r )powersrrc3:># UHnTUTU-v M g7frFrb)rrr alpha_powerss rr,_prime_decomp_split_ideal..s;]qtLO#]s)r9 to_parentmodulerrrquogcdexr|r}rrrepto_listrrrrrrbasis_elementscolumn columnspacerappend)rkr"rrr!r#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 *  $ $Q 'q N 0 s %G= cFUcUc [S5eUb [U5 UcURRn0nUbUc [ XS9up#UR 5nXc-nXp-S:wa [ X5$U=(d# URU5=(d [X 5nU/n/n U(avUR5n [XU5upU RS:Xa[XU5n U RU 5 O![XXU5upURX/5 U(aMvU $)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.)radicalsrr ) ValueErrorrr9r:r discriminantrgetr poprr5rrrextend)r"r:r!dKradicalrdT f_squaredstackprimesrkrrrI1I2s r prime_decompr sn yRZDEE ~226y IIKKH zRZ10  BI}&q--CaC,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.polyutilsr sympy.utilities.decoratorr basisr r exceptionsrmodulesrr utilitiesrrrrrfrjrrrrrr rbrrrs%&.0.:12,.&6:,0yB!yBx +\U U pBJ&2j':'TOOr