\hPNSrSSKJr SSKrSSKJrJr SSKJr SSK J r SSK J r SSK Jr SS KJr SS KJrJr SS KJrJrJrJr SS KJr SS KJrJrJrJr SSK J!r! SSK"J#r# "SS\5r$SSjr%Sr&SSjr'SSjr(SSjr)SSjr*SSjr+SSjr,\#SSSS.Sj5r-g) 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\rSrSrSrg)MaxTriesException#N)__name__ __module__ __qualname____firstlineno____static_attributes__r]/var/www/auris/envauris/lib/python3.13/site-packages/sympy/polys/numberfields/galoisgroups.pyrr#sr!rcl^[S5nUR5nUc [5nURUR5 U(a0mSnSnU4Sjn [ U5GHCn U(aU "W5n [ U 5n [SU 55n X-W:a)US:Xa US- nUS- nOUS-nU "U5n [ U 5n [S5/U Vs/sHn[U5PM sn-nO?[U S-S-U5n[R"SUS- 5n[UU*U[5n[XR5n[URUU- 5U5nURU;dGM[!URR#5[5(dGM@UU4s $ [$es snf)a Given a univariate, monic, irreducible polynomial over the integers, find another such polynomial defining the same number field. Explanation =========== See Alg 6.3.4 of [1]. Parameters ========== T : Poly The given polynomial max_coeff : int When choosing a transformation as part of the process, keep the coeffs between plus and minus this. max_tries : int Consider at most this many transformations. history : set, None, optional (default=None) Pass a set of ``Poly.rep``'s in order to prevent any of these polynomials from being returned as the polynomial ``U`` i.e. the transformation of the given polynomial *T*. The given poly *T* will automatically be added to this set, before we try to find a new one. fixed_order : bool, default True If ``True``, work through candidate transformations A(x) in a fixed order, from small coeffs to large, resulting in deterministic behavior. If ``False``, the A(x) are chosen randomly, while still working our way up from small coefficients to larger ones. Returns ======= Pair ``(A, U)`` ``A`` and ``U`` are ``Poly``, ``A`` is the transformation, and ``U`` is the transformed polynomial that defines the same number field as *T*. The polynomial ``A`` maps the roots of *T* to the roots of ``U``. Raises ====== MaxTriesException if could not find a polynomial before exceeding *max_tries*. XcJ>TRU[US55nUTU'U$)N)getr)degreegencoeff_generatorss r"get_coeff_generator9tschirnhausen_transformation..get_coeff_generatorcs,""6<+BC#&  r!c38# UHn[U5v M g7f)N)abs).0cs r" /tschirnhausen_transformation..xs+FqCFFFsr))rr+setaddreprangenextmaxrminrandomrandintrrr, resultantrto_listr)T max_coeff max_trieshistory fixed_orderr%n deg_coeff_sumcurrent_degreer.ir,coeffsmr3aCdAUr-s @r"tschirnhausen_transformationrR'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 2sF1c[U[5(aUR5O[U[5n[ U5$)z?Convenience to check if a Poly or dup has square discriminant. ) isinstancer discriminantr rr)rBrOs r"has_square_discrVs.&q$//5Ea5LA Q<r!FcfSSKJn [U5(aURS4$URS4$)zj Compute the Galois group of a polynomial of degree 3. Explanation =========== Uses Prop 6.3.5 of [1]. r)S3TransitiveSubgroupsTF)sympy.combinatorics.galoisrXrVA3S3)rBrD randomizerXs r"_galois_group_degree_3r]s9A0?0B0B " % %t ,4'**E24r!c SSKJn SSKJn [ S5nUSUS-USUS--nU"S5U"S5"SS5U"S5"SS5/n[ XeU5nUSUSS--USUSS---USUSS---USUSS---n U"S5U"S5"SS5/n [ 5n [U5GH6n U S:a[XU U(+S9upURUS S 9upn[U[5(dMF[U5nUc'U(aURS 4s $URS 4s $U(aURS 4s $XnU R![#UU"U55S S 9nU Vs/sH nUU-U-PM nn[ UUU5nURU5un n [%U[5nUS:XaGM ['U5(aUR(S 4s $UR*S 4s $ [,es snf) 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. r PermutationS4TransitiveSubgroupsz X0 X1 X2 X3r'r)r&rDrErFT)find_integer_rootF simultaneous) sympy.combinatorics.permutationsr`rYrbrrr7r:rR eval_for_polyrrrVA4S4Vsubszipr rC4D4r)rBrDr\r`rbr%F1s1R1F2_pres2_prerErJ_R_dupi0sq_discsigmaF2taus2R2rOs r""_galois_group_degree_4_root_approxr~sN=@ 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#eCio 0 r1b !&&q) q! UB ' 6  Q<<),,e4 4),,e4 4]` 1sHc PSSKJn [5n[U5H8n[ US5n[ U[ 5(a O[XUU(+S9uppM: [e[U[ 5n[[USV V s/sHup[U 5S- /U -PM sn n /55n U S/:Xa,[U5(aURS4$URS4$U /SQ:XaUR S4$U /S Q:XaUR"S4$U S S /:XdeUR$S4$s sn n f) 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. rrarcr)TFr)r))r'r'r'r'r)rYrbr7r:rrrrRrr sortedsumlenrVrirjrnrkro) rBrDr\rbrErJrvruflreLs r"_galois_group_degree_4_lookuprs=AeG 9 '1- UB   +A4;@-11599 $<S 4$UR@S4$U(aURBS 4$URDS4$s sn n f)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. r)S6TransitiveSubgroupsr)rc)r)r'r&r&Fr'rTrr6)r)r)r)r&r)#rYrr7r:rrrrRrr rlistrappendrrrrVC6D6G18G36mS4pA4xC2S4xC2riS4mPSL2F5PGL2F5r[r A6S6G36pG72)rBrDr\rrErJrvrurfactors_by_degrrOffr T_has_sq_discf1f2 any_squares r"_galois_group_degree_6_lookuprsFAeG 9 '1- UB   +A4;>> 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_groupr)Nr)rrrr) frrDr\gensargsFoptexcs r"rrspD :2D :2D81D1D1 >>'$-  // 83778s9 A AA) r#NT)r#F).__doc__ collectionsrr>sympy.core.symbolrrsympy.ntheory.primetestrsympy.polys.domainsrsympy.polys.densebasicrsympy.polys.densetoolsr sympy.polys.euclidtoolsr sympy.polys.factortoolsr r *sympy.polys.numberfields.galois_resolventsr rrr"sympy.polys.numberfields.utilitiesrsympy.polys.polytoolsrrrrsympy.polys.sqfreetoolsrsympy.utilitiesrrrRrVr]r~rrrrrrr!r"rs $ ,-"-+4F<JJ-",IM-1hV 4Tn(-VNb*0ZZ9z#(B%j/j/r!