o o h]@sdZddlmZddlmZddlmZddlmZddl m Z ddl m Z ddl mZdd lmZd d lmZmZd d lmZd d lmZmZd dlmZmZddZGdddeZddZeddZd$ddZ ddZ!ddZ"ddZ#d d!Z$ed%d"d#Z%dS)&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_subspacecCsHd}d}|s d}n |sd}n|sd}|dur"t||dS)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 ]) rparentTgenrrr"rstr numeratorr)denom) r# field_gen just_gensrrr"r alpha_repgensrrrreprUs! zPrimeIdeal.reprcCs|SN)r9r*rrr__repr__szPrimeIdeal.__repr__cCs(|j|j|j|j}d|_d|_|S)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#Mrrr as_submodules7zPrimeIdeal.as_submodulecCst|tr ||kStSr:) isinstancerr?NotImplementedr#otherrrr__eq__s zPrimeIdeal.__eq__cCs ||S)z Convert to a :py:class:`~.Submodule` and add to another :py:class:`~.Submodule`. See Also ======== as_submodule r?rBrrr__add__ zPrimeIdeal.__add__cCs ||S)z Convert to a :py:class:`~.Submodule` and multiply by another :py:class:`~.Submodule` or a rational number. See Also ======== as_submodule rErBrrr__mul__rGzPrimeIdeal.__mul__cCs|jSr:)rr*rrr _zeroth_powerszPrimeIdeal._zeroth_powercCs|Sr:rr*rrr _first_powerszPrimeIdeal._first_powercCs&|jdurt|j|jg|j|_|jS)aO 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. N)r _compute_test_factorrrrr*rrr test_factors zPrimeIdeal.test_factorcCs t||S)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#Irrrr!s zPrimeIdeal.valuationcCs||S)a 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 )r?reduce_element)r#eltrrrrOszPrimeIdeal.reduce_elementcCs |jj|}||}|S)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 )rr/element_from_ANPrOto_ANPr#arPredrrr reduce_ANP*s zPrimeIdeal.reduce_ANPcCs0|jj|}||}|tt|jS)a 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 ) rr/element_from_alg_numrO field_elementlistreversedQQ_colflatrSrrrreduce_alg_numGs zPrimeIdeal.reduce_alg_numr:)NF)__name__ __module__ __qualname____doc__r$r+propertyr(r9r;r?rDrF__radd__rH__rmul__rIrJrLr!rOrVr]rrrrr)s*   +=   rcsxt||fdd|D}td|jftj|}|dddf}|j |j | t |j d}|S)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.) csg|] }|jdqS)modulus)inner_endomorphismmatrix.0gErrr sz(_compute_test_factor..rNr4)rendomorphism_ringrzerosr,rvstack nullspace transposer/rh convert_torr4)rr8rmatricesBr.betarrlrrKesrKcCsZ|j|j}}|j|j|j}}}|t|j||j}|t}| }||dkr2dS| } ||| } | |dk} d} ||}t |D])} |j |dd| f|d}|| 9}| |}t |D] }||||| f<qmqO||d|dfj|dkr | S||}| rz|t}WntyY| Sw|t}| d7} qG)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.) rTNror )rrr,rhr4rurinvrdetrLranger/ representr\elementr)rNPrrr,WdADrxrneed_complete_testvjcirrrrMsD*       rMNc st||j}|j}tfdd|Dr|S|dur0|dur&|}n t||j}| }fdd|ddD}||7}t t |d} | D]!} t ddt | |D} | ||} | dkro| SqNdS) 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.) c3s|] }|dVqdS)rN)equivrirrr sz_two_elt_rep..Ncsg|]}|qSrrrjomrrrrn#sz _two_elt_rep..r css|] \}}||VqdSr:r)rjcibetairrrr'sr)rr/r0allzeroabssubmodule_from_gensrhrzbasis_element_pullbacksrlensumzipnorm) r8rrrNppbr0omegarxsearchrrr,rrr _two_elt_reps((   rcsjjj}t|d}|\}}t|dkr+|dddkr+tjjdgSfdd|DS)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``. rer rc s4g|]\}}tjt|td||qSdomain)rr/element_from_polyrrdegree)rjtr"rrrrrn>s z+_prime_decomp_easy_case..)r/r0r factor_listrrrr,)rrr0T_barlcflrrr_prime_decomp_easy_case1s   rc s|j}|j\}}|dkr||t}n|||tdddf}|jd|kr5t|tt}||}| | |}t |fdd}|j d} | sXJ| |fS)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 cs ||Sr:rr-rrrss z._prime_decomp_compute_kernel..re)rhshapeeyerhstackrrursubmodule_from_matrixcompute_mult_tabdiscard_beforerkernelr) rNrrrr,rrwGphiNrrr_prime_decomp_compute_kernelDs     rcs\|jj\}}||}j|jfddtjdD}t|||d}t|||S)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 cs(g|]}jdd|fjdqS)Nro)r/r4)rjrrrrrrns(z/_prime_decomp_maximal_ideal..r )r)rhrr{rr)rNrrmr,rr8rrrr_prime_decomp_maximal_idealys  rcsR|j|kr|j|ur|j|usJ|d}|j|usJgt|td}|\}}|dd} || } | | \} } } | dksHJtt t | | t dj tfddttD}d|}||g}g}|D]3}|j|usJ|jtjfdd|D}|t }||}||qs|S) 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 )powersrrc3s |] }||VqdSr:r)rjr)rm alpha_powersrrrsz,_prime_decomp_split_ideal..cs g|] }|jtdqSr)columnrr)r"rrrrnsz-_prime_decomp_split_ideal..)r/ to_parentmodulerrrquogcdexrYrZrrrepto_listrr{rrhrurbasis_elements columnspacerappend)rNrrrrrrrrm1m2UVrkeps1eps2idempsfactorsepsrrHr)rmrr"rr_prime_decomp_split_ideals2"        rcCs |dur |dur td|durt||dur|jj}i}|dus&|dur.t||d\}}|}||}||dkrAt||S|pL||pLt||}|g}g} |r| } t | ||\} } | j dkrqt | ||} | | nt| || | |\}}|||g|sT| S)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.) Nz)At least one of T or ZK must be provided.)radicalsrr ) ValueErrorrr/r0r discriminantrgetr poprr,rrrextend)rr0rdKradicalrdT f_squaredstackprimesrNrrr~I1I2rrr prime_decomps47      r)NN)NNNN)&rasympy.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 modulesrr utilitiesrrrrrKrMrrrrrrrrrrs6         >.  XE5*