+ iPPRt^RIHt^RIt^RIHtHt^RIHt^RI H t ^RI H t ^RI Ht^RIHt^R IHtHt^R IHtHtHtHt^R IHt^R IHtHtHtHt^R I H!t!^RI"H#t#!RR]4t$RRlt%Rt&RRlt'RRlt(RRlt)RRlt*RRlt+RRlt,]#RRR^RR/Rl4t-R#) z Compute Galois groups of polynomials. We use algorithms from [1], with some modifications to use lookup tables for resolvents. References ========== .. [1] Cohen, H. *A Course in Computational Algebraic Number Theory*. ) defaultdictN)Dummysymbols) is_square)ZZ) dup_random)dup_eval)dup_discriminant)dup_factor_listdup_irreducible_p)GaloisGroupExceptionget_resolvent_by_lookupdefine_resolvents Resolvent) coeff_search)Polypoly_from_exprPolificationFailedComputationFailed) dup_sqf_p)publicc]tRt^#tRtR#)MaxTriesExceptionN)__name__ __module__ __qualname____firstlineno____static_attributes__rڅ/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/polys/numberfields/galoisgroups.pyrr#srrca\R4pVP4pVf \4pVPVP4V'd/o^p^pV3Rlp \ V4EFkp V'dV !X4p \ V 4p \RV 44p W,X8d7V^8XdV^, pV^, pM V^,pV !V4p \ V 4p \^4.V Uu.uFp\V4NK up,pMK\V ^,^,V4p\P!^V^, 4p\VV)V\4p\WP4p\VPVV, 4V4pVPV9gEK7\!VPP#4\4'gEKhVV3u# \$huupi)a Given a univariate, monic, irreducible polynomial over the integers, find another such polynomial defining the same number field. Explanation =========== See Alg 6.3.4 of [1]. Parameters ========== T : Poly The given polynomial max_coeff : int When choosing a transformation as part of the process, keep the coeffs between plus and minus this. max_tries : int Consider at most this many transformations. history : set, None, optional (default=None) Pass a set of ``Poly.rep``'s in order to prevent any of these polynomials from being returned as the polynomial ``U`` i.e. the transformation of the given polynomial *T*. The given poly *T* will automatically be added to this set, before we try to find a new one. fixed_order : bool, default True If ``True``, work through candidate transformations A(x) in a fixed order, from small coeffs to large, resulting in deterministic behavior. If ``False``, the A(x) are chosen randomly, while still working our way up from small coefficients to larger ones. Returns ======= Pair ``(A, U)`` ``A`` and ``U`` are ``Poly``, ``A`` is the transformation, and ``U`` is the transformed polynomial that defines the same number field as *T*. The polynomial ``A`` maps the roots of *T* to the roots of ``U``. Raises ====== MaxTriesException if could not find a polynomial before exceeding *max_tries*. XcJ<SPV\V^44pVSV&V#))getr)degreegencoeff_generatorss& r get_coeff_generator9tschirnhausen_transformation..get_coeff_generatorcs,""6<+BC#&  rc38"TFp\V4xK R#5i)N)abs).0cs& r /tschirnhausen_transformation..xs+FqCFFFs)rr&setaddreprangenextmaxrminrandomrandintrrr' resultantrto_listr)T max_coeff max_trieshistory fixed_orderr"n deg_coeff_sumcurrent_degreer)ir'coeffsmr.aCdAUr(s&&&&& @r tschirnhausen_transformationrL'sb c A  A% KK  9   &n5C#YF+F++A!M1!Q&!Q&M%2Q%6N"a'N).9cA&1&Q2a5&11A AqD1Hi(Aq!a%(A1qb!R(A EEN QU#Q ' 55 Iaeemmor$B$Ba4KMN 2s(Gc\V\4'dVP4M\V\4p\ V4#)z?Convenience to check if a Poly or dup has square discriminant. ) isinstancer discriminantr rr)r<rIs& r has_square_discrPs.&q$//5Ea5LA Q<rFch^RIHp\V4'dVPR3#VPR3#)zj Compute the Galois group of a polynomial of degree 3. Explanation =========== Uses Prop 6.3.5 of [1]. )S3TransitiveSubgroupsTF)sympy.combinatorics.galoisrRrPA3S3)r<r> randomizerRs&&& r _galois_group_degree_3rWs9A0?0B0B " % %t ,4'**E24rc  ^RIHp^RIHp\ R4pV^,V^,,V^,V^,,,pV!^4V!^4!^^4V!^4!^^4.p\ WeV4pV^,V^,^,,V^,V^,^,,,V^,V^,^,,,V^,V^,^,,,p V!^4V!^4!^^4.p \ 4p \V4EFQp V ^8d\WV V'*R7wrVPVRR7wrp\V\4'gKN\V4pVf)V'dVPR3u#VPR3u#V'dVPR3u#W,pV P!\#VV!V44RR7pV Uu.uFpVV,V,NK pp\ VVV4pVPV4wpp \%V\4pV^8XdEK#\'V4'dVP(R3u#VP*R3u# \,huupi) z Compute the Galois group of a polynomial of degree 4. Explanation =========== Follows Alg 6.3.7 of [1], using a pure root approximation approach.  PermutationS4TransitiveSubgroupsz X0 X1 X2 X3r>r?r@T)find_integer_rootF simultaneous) sympy.combinatorics.permutationsrZrSr\rrr1r4rL eval_for_polyrrrPA4S4Vsubszipr rC4D4r)r<r>rVrZr\r"F1s1R1F2_pres2_prer?rD_R_dupi0sq_discsigmaF2taus2R2rIs&&& r "_galois_group_degree_4_root_approxrxsP=@ A 1adQqT!A$Y BAAq!Aq! B 2" B qT!A$'\AaD1qL (1Q4!a< 7!A$qtQw, FFAAq!F eG 9  q5/8?@IMKDA''T'B "## "!$ :9@*--t4 </22E: < )++T2 2 [[Qa)[ =)/ 0#eCioo 0 r1b !&&q) q! UB ' 6  Q<<),,e4 4),,e4 4]` 1s;J c x^RIHp\4p\V4F<p\ V^4p\ V\ 4'dM\WVV'*R7wrpK> \h\V\ 4p\\V^,U U u.uFwr\V 4^, .V ,NK! up p .44p V ^.8Xd.\V4'dVPR3#VPR3#V .RO8XdVP R3#V .RO8XdVP"R3#V ^^.8XgQhVP$R3#uup p i)z Compute the Galois group of a polynomial of degree 4. Explanation =========== Based on Alg 6.3.6 of [1], but uses resolvent coeff lookup. r[r]TFr$r$)r|r|)rSr\r1r4r rrrLrr sortedsumlenrPrcrdrhreri) r<r>rVr\r?rDrproflreLs &&& r _galois_group_degree_4_lookuprs@AeG 9 '1- UB   +A4;rVrrZX5resF51ros51R51r?reached_second_stagerDR51_duprr rounded_rootspermutation_indexcandidate_rootr"rmrnrqrsrtrurvrwrprIs&&& r _galois_group_degree_5_hybridr)sA< ! "B  Cf+KCC ++r C CS !CeG  9  q5/8?@IMKDA*!Q/"%%  $%a(G "--@-11599 $< \h\V4p \V\ 4'd%V 'dVPR3#VPR3#V 'gVPR3#\V\ P!V4R7P!4^,p \#V 4^8XdVP$R3#VP&R3#)z Compute the Galois group of a polynomial of degree 5. Explanation =========== Based on Alg 6.3.9 of [1], but uses resolvent coeff lookup, plus factorization over an algebraic extension. rr]TF)domain)rSrr1r4r rrrLrrPr rrrralg_field_from_poly factor_listrrr) r<r>rVr_Tr?rDrprorrrs &&& r (_galois_group_degree_5_lookup_ext_factorrzs A BeG 9 '1- UB   +A4; \h\V\ 4p\\4p V^,F-wrV \V 4^, ,PV 4K/ \\V P!4U U u.uFwrV .\V 4,NK up p .44p \#V4pV .RO8Xd>V ^,^,p\#V4'dVP$R3#VP&R3#V ^^.8XdOV ^,wpp\#V4;'g \#V4pV'dVP(R3#VP*R3#V ^^.8XdTV'dVP,R3#V ^,^,p\#V4'dVP.R3#VP0R3#V .RO8Xd%V'dVP2R3#VP4R3#V ^^.8Xd%V'dVP6R3#VP8R3#V .RO8XdVP:R3#V ^.8XgQh\4p\V4F<p\ V^4p\ V\ 4'dM\WVV'*R7wrpK> \h\#V4p\=V\ 4'd%V'dVP>R3#VP@R3#V'dVPBR3#VPDR3#uup p i)z Compute the Galois group of a polynomial of degree 6. Explanation =========== Based on Alg 6.3.10 of [1], but uses resolvent coeff lookup. )S6TransitiveSubgroupsr]FT)r$r|rz)r$r$r$r)#rSrr1r4r rrrLrr rlistrappendr}r~rrPC6D6G18G36mS4pA4xC2S4xC2rcS4mPSL2F5PGL2F5rUr A6S6G36pG72)r<r>rVrr?rDrprorfactors_by_degrrIffr T_has_sq_discf1f2 any_squares&&& r _galois_group_degree_6_lookuprsMAeG 9 '1- UB   +A4;rVcT;'g.pT;'g/p\V.VO5/VBwrgTPYTR7# \dp\R^T4hRp?ii;i)al Compute the Galois group for polynomials *f* up to degree 6. Examples ======== >>> from sympy import galois_group >>> from sympy.abc import x >>> f = x**4 + 1 >>> G, alt = galois_group(f) >>> print(G) PermutationGroup([ (0 1)(2 3), (0 2)(1 3)]) The group is returned along with a boolean, indicating whether it is contained in the alternating group $A_n$, where $n$ is the degree of *T*. Along with other group properties, this can help determine which group it is: >>> alt True >>> G.order() 4 Alternatively, the group can be returned by name: >>> G_name, _ = galois_group(f, by_name=True) >>> print(G_name) S4TransitiveSubgroups.V The group itself can then be obtained by calling the name's ``get_perm_group()`` method: >>> G_name.get_perm_group() PermutationGroup([ (0 1)(2 3), (0 2)(1 3)]) Group names are values of the enum classes :py:class:`sympy.combinatorics.galois.S1TransitiveSubgroups`, :py:class:`sympy.combinatorics.galois.S2TransitiveSubgroups`, etc. Parameters ========== f : Expr Irreducible polynomial over :ref:`ZZ` or :ref:`QQ`, whose Galois group is to be determined. gens : optional list of symbols For converting *f* to Poly, and will be passed on to the :py:func:`~.poly_from_expr` function. by_name : bool, default False If ``True``, the Galois group will be returned by name. Otherwise it will be returned as a :py:class:`~.PermutationGroup`. max_tries : int, default 30 Make at most this many attempts in those steps that involve generating Tschirnhausen transformations. randomize : bool, default False If ``True``, then use random coefficients when generating Tschirnhausen transformations. Otherwise try transformations in a fixed order. Both approaches start with small coefficients and degrees and work upward. args : optional For converting *f* to Poly, and will be passed on to the :py:func:`~.poly_from_expr` function. Returns ======= Pair ``(G, alt)`` The first element ``G`` indicates the Galois group. It is an instance of one of the :py:class:`sympy.combinatorics.galois.S1TransitiveSubgroups` :py:class:`sympy.combinatorics.galois.S2TransitiveSubgroups`, etc. enum classes if *by_name* was ``True``, and a :py:class:`~.PermutationGroup` if ``False``. The second element is a boolean, saying whether the group is contained in the alternating group $A_n$ ($n$ the degree of *T*). Raises ====== ValueError if *f* is of an unsupported degree. MaxTriesException if could not complete before exceeding *max_tries* in those steps that involve generating Tschirnhausen transformations. See Also ======== .Poly.galois_group galois_groupN)rr>rV)rrrr) frr>rVgensargsFoptexcs &$$$*, r rrstD ::2D ::2D81D1D1 >>'$-  // 83778s> A AA) NT)rF).__doc__ collectionsrr8sympy.core.symbolrrsympy.ntheory.primetestrsympy.polys.domainsrsympy.polys.densebasicrsympy.polys.densetoolsrsympy.polys.euclidtoolsr sympy.polys.factortoolsr r *sympy.polys.numberfields.galois_resolventsr r rr"sympy.polys.numberfields.utilitiesrsympy.polys.polytoolsrrrrsympy.polys.sqfreetoolsrsympy.utilitiesrrrLrPrWrxrrrrrrrr rs $ ,-"-+4F<JJ-",hV 4Tn(-VNb*0ZZ9zj/5j/Bj/%j/j/r