a =$ 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|SNrrqrr Kz"nilradical_mod_p..r )nrkernel)Hrr%r(phirr$r nilradical_mod_p%s! r,c st|||d}|jj|j|j|jd}|||}|t|fdd}|j|d}|jj|j|j|j|d}||} | |fS)zD Perform the second enlargement in the Round Two algorithm. r$)denomcs |Sr")Zinner_endomorphismr#Err r&Wr'z%_second_enlargement..r )r,parentZsubmodule_from_matrixmatrixr-Zendomorphism_ringr r)) r*rr%ZIpBCr+gammaGH1rr.r _second_enlargementOs  r7cCs|jtkr.zt|td}Wnty,Yn0|jrJ|jrJ|jrJ|jtksRtd| }| }t t |}t |\}}t|}|}d} |r6|\} } t|| \} } | dkrq|t| td}|j|| ||d}| | krq| }||kr|| 9}qt|| |\}} ||kr|}t|| |\}} qq| durTt|trT| || <|}d|_d|_||jd|jd|}||fS)a Zassenhaus's "Round 2" algorithm. Explanation =========== Carry out Zassenhaus's "Round 2" algorithm on a monic irreducible polynomial *T* over :ref:`ZZ`. This computes an integral basis and the discriminant for the field $K = \mathbb{Q}[x]/(T(x))$. 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, theta >>> T = Poly(x ** 3 + x ** 2 - 2 * x + 8) >>> K = QQ.algebraic_field((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**2/2 + theta/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` The irreducible monic polynomial over :ref:`ZZ` defining the number field. 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.* rzCRound 2 requires a monic irreducible univariate polynomial over ZZ.Nr)Z hnf_modulusT)rrrrrZ is_univariateZis_irreducibleZis_monic ValueErrorrZ discriminantZ from_sympyabsr r Zwhole_submodulepopitemr!Zelement_from_polyaddr7 isinstancedictZ_starts_with_unityZ_is_sq_maxrank_HNFr1Zdetr-)rZradicalsr(DZ D_modulusrFZZthetar*ZnilradrerrUr%r6ZZKZdKrrr round_two^sV[       rC)N)N)__doc__Zsympy.polys.polytoolsrZsympy.polys.domains.integerringrZ!sympy.polys.domains.rationalfieldrZsympy.polys.polyerrorsrZsympy.utilities.decoratorrmodulesrr r Z utilitiesr r!r,r7rCrrrr s       *