+ iE Rt^RIHt^RIHt^RIHt^RIHt^RI H t ^RI H t H t Ht^RIHtRtR R ltR t] R R l4tR #)z,Computing integral bases for number fields. )Poly)AlgebraicField)ZZ)QQ)public)ModuleEndomorphismModuleHomomorphism PowerBasis) extract_fundamental_discriminantcVPp\WR7pVP4wrEV^8XgQh\^W!R7pVF wrxWg,pK W6,p \V\R7p \V \R7p W,V, V,p \WR7p T pWi3FpVP V4pK W>,pVP 4pVV3#)zn Apply the "Dedekind criterion" to test whether the order needs to be enlarged relative to a given prime *p*. modulusdomain)genr factor_listrgcddegree)TpxT_barlcflg_barti_bar_h_barghff_barZ_barbU_barms&& ~/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/polys/numberfields/basis.py_apply_Dedekind_criterionr' s A  E    FB 7N7 A !E   NE U2A U2A qA  E E^ !  NE A !8ONcaVPpSfVoSV8d SV,oK\VV3Rl4pVPVR7#)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.) c<VS,#N)rqs&r&"nilradical_mod_p..Ks !Q$r(r )nrkernel)Hrr-r0phis&&f r&nilradical_mod_pr4%sFB Ay !e FA Q /C ::a:  r(ca \WVR7pVPPVPVP,VPR7pWAV,,pVP 4o \ VS V 3Rl4pVPVR7pVPPVPVP,VPV,R7pW,p W3#)z< Perform the second enlargement in the Round Two algorithm. )r-)denomc&<SPV4#r+)inner_endomorphism)rEs&r&r.%_second_enlargement..WsQ-A-A!-Dr(r )r4parentsubmodule_from_matrixmatrixr6endomorphism_ringrr1) r2rr-IpBCr3gammaGH1r9s &&& @r&_second_enlargementrEOs !! $B &&qxx"))';177&KA aCA A Q#D EC JJqJ !E &&qxx%,,'>aggPQk&RA B 6Mr(cdRp\V\4'dYPP4rVP'd.VP 'dVP \\39d \R4hVP4wrVP4pVP4p\P!\V44p\V4wr7\!T;'gT4pVP#4p Rp V'dVP%4wr\'W 4wrV^8XdK0VP)\+V \R74pV P-W,V ,VR7p W8:dKxT pVV8d VV ,pK\/WV4wpp VV 8wgKTp \/WV4wpp KV e\V\04'dWX &T pRVnRVnVVP6P94^,,VP:^V,,,pVV3#)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 ========== .. 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