U ;qLeE @sdZddlmZddlmZddlmZddlmZddl m Z ddl m Z m Z mZdd lmZd d Zdd dZddZe dddZd S)z,Computing integral bases for number fields. )Poly)AlgebraicField)ZZ)QQ)public)ModuleEndomorphismModuleHomomorphism PowerBasis) extract_fundamental_discriminantcCs|j}t||d}|\}}|dks*ttd||d}|D]\}}||9}q<||} t|td} t| td} | | ||} t| |d} | }|| fD]}||}q||}|}||fS)zz Apply the "Dedekind criterion" to test whether the order needs to be enlarged relative to a given prime *p*. modulusrdomain)genr factor_listAssertionErrorrgcddegree)TpxZT_barlcflZg_barZti_bar_Zh_barghfZf_barZZ_barbU_barmr!u/home/p21-0144/sympy/latex2sympy2solve-back-end/sympyEq/lib/python3.8/site-packages/sympy/polys/numberfields/basis.py_apply_Dedekind_criterion s$          r#NcsB|j}dkr$||kr$|9qt|fdd}|j|dS)a Compute the nilradical mod *p* for a given order *H*, and prime *p*. Explanation =========== This is the ideal $I$ in $H/pH$ consisting of all elements some positive power of which is zero in this quotient ring, i.e. is a multiple of *p*. Parameters ========== H : :py:class:`~.Submodule` The given order. p : int The rational prime. q : int, optional If known, the smallest power of *p* that is $>=$ the dimension of *H*. If not provided, we compute it here. Returns ======= :py:class:`~.Module` representing the nilradical mod *p* in *H*. References ========== .. [1] Cohen, H. *A Course in Computational Algebraic Number Theory*. (See Lemma 6.1.6.) Ncs|SNr!rqr!r"Kz"nilradical_mod_p..r )nrkernel)Hrr'r*phir!r&r"nilradical_mod_p%s! r.c st|||d}|jj|j|j|jd}|||}|t|fdd}|j|d}|jj|j|j|j|d}||} | |fS)zD Perform the second enlargement in the Round Two algorithm. r&)denomcs |Sr$)Zinner_endomorphismr%Er!r"r(Wr)z%_second_enlargement..r )r.parentZsubmodule_from_matrixmatrixr/Zendomorphism_ringr r+) r,rr'ZIpBCr-gammaGH1r!r0r"_second_enlargementOs  r9cCsd}t|tr||j}}|jr8|jr8|jttfkr@t d| \}}| }| }t t|}t|\}}t|p~|}|} d} |r4|\} } t|| \} }|dkrq|t| td}| j|| | |d} | |krq| }||kr|| 9}qt| | |\}} || kr|} t| | |\}} qq| dk rRt|trR| || <| }d|_d|_||jd|jd|}||fS)a Zassenhaus's "Round 2" algorithm. Explanation =========== Carry out Zassenhaus's "Round 2" algorithm on an irreducible polynomial *T* over :ref:`ZZ` or :ref:`QQ`. This computes an integral basis and the discriminant for the field $K = \mathbb{Q}[x]/(T(x))$. Alternatively, you may pass an :py:class:`~.AlgebraicField` instance, in place of the polynomial *T*, in which case the algorithm is applied to the minimal polynomial for the field's primitive element. Ordinarily this function need not be called directly, as one can instead access the :py:meth:`~.AlgebraicField.maximal_order`, :py:meth:`~.AlgebraicField.integral_basis`, and :py:meth:`~.AlgebraicField.discriminant` methods of an :py:class:`~.AlgebraicField`. Examples ======== Working through an AlgebraicField: >>> from sympy import Poly, QQ >>> from sympy.abc import x >>> T = Poly(x ** 3 + x ** 2 - 2 * x + 8) >>> K = QQ.alg_field_from_poly(T, "theta") >>> print(K.maximal_order()) Submodule[[2, 0, 0], [0, 2, 0], [0, 1, 1]]/2 >>> print(K.discriminant()) -503 >>> print(K.integral_basis(fmt='sympy')) [1, theta, theta/2 + theta**2/2] Calling directly: >>> from sympy import Poly >>> from sympy.abc import x >>> from sympy.polys.numberfields.basis import round_two >>> T = Poly(x ** 3 + x ** 2 - 2 * x + 8) >>> print(round_two(T)) (Submodule[[2, 0, 0], [0, 2, 0], [0, 1, 1]]/2, -503) The nilradicals mod $p$ that are sometimes computed during the Round Two algorithm may be useful in further calculations. Pass a dictionary under `radicals` to receive these: >>> T = Poly(x**3 + 3*x**2 + 5) >>> rad = {} >>> ZK, dK = round_two(T, radicals=rad) >>> print(rad) {3: Submodule[[-1, 1, 0], [-1, 0, 1]]} Parameters ========== T : :py:class:`~.Poly`, :py:class:`~.AlgebraicField` Either (1) the irreducible polynomial over :ref:`ZZ` or :ref:`QQ` defining the number field, or (2) an :py:class:`~.AlgebraicField` representing the number field itself. radicals : dict, optional This is a way for any $p$-radicals (if computed) to be returned by reference. If desired, pass an empty dictionary. If the algorithm reaches the point where it computes the nilradical mod $p$ of the ring of integers $Z_K$, then an $\mathbb{F}_p$-basis for this ideal will be stored in this dictionary under the key ``p``. This can be useful for other algorithms, such as prime decomposition. Returns ======= Pair ``(ZK, dK)``, where: ``ZK`` is a :py:class:`~sympy.polys.numberfields.modules.Submodule` representing the maximal order. ``dK`` is the discriminant of the field $K = \mathbb{Q}[x]/(T(x))$. See Also ======== .AlgebraicField.maximal_order .AlgebraicField.integral_basis .AlgebraicField.discriminant References ========== .. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.* NzDRound 2 requires an irreducible univariate polynomial over ZZ or QQ.rr)Z hnf_modulusT) isinstancerextminpoly_of_element is_univariateis_irreduciblerrr ValueError)make_monic_over_integers_by_scaling_rootsr discriminant from_sympyabsr r Zwhole_submodulepopitemr#Zelement_from_polyraddr9dictZ_starts_with_unityZ_is_sq_maxrank_HNFr3detr/)rradicalsKrr*DZ D_modulusFZZthetar,Znilradrerr Ur'r8ZKdKr!r!r" round_two^sP`         rQ)N)N)__doc__sympy.polys.polytoolsrZ"sympy.polys.domains.algebraicfieldrZsympy.polys.domains.integerringrZ!sympy.polys.domains.rationalfieldrsympy.utilities.decoratorrmodulesrr r utilitiesr r#r.r9rQr!r!r!r"s       *