[{"data":1,"prerenderedAt":1272},["ShallowReactive",2],{"subject:abstract-algebra":3,"course-wordcounts":75,"nav:abstract-algebra":987},{"id":4,"title":5,"blurb":6,"body":7,"brief":18,"category":56,"description":57,"draft":58,"extension":59,"meta":60,"module":15,"navigation":25,"path":61,"practice":62,"rawbody":63,"readingTime":64,"seo":69,"sources":70,"status":71,"stem":72,"summary":15,"topics":73,"__hash__":74},"course\u002F05.abstract-algebra\u002Findex.md","Abstract Algebra","Groups, rings, and fields — the axioms behind symmetry and arithmetic, from\npermutations and Sylow theorems to Galois theory and why the quintic has no\nformula.\n",{"type":8,"value":9,"toc":14},"minimark",[10],[11,12,13],"p",{},"Two ideas run through the whole subject: an operation, and the symmetry it\nencodes. The sequence starts with groups — their axioms, the cyclic and\ndihedral examples, permutations and the symmetric group — then develops the\nmachinery that makes them tractable: subgroups and cosets, Lagrange's theorem,\nnormal subgroups and quotients, homomorphisms and the isomorphism theorems, and\nthe Sylow theorems that dissect finite groups by prime. From there it adds a\nsecond operation to reach rings (ideals, quotients, polynomial and Euclidean\ndomains, unique factorization) and modules (the structure theorem that classifies\nthem over a PID, with linear algebra and finitely generated abelian groups as\ncorollaries). The last movement is field theory and Galois theory, connecting\nfield extensions back to groups and resolving the classical impossibilities —\nthe insolvability of the quintic, the constructions that compass and straightedge\ncannot reach. These notes follow Dummit & Foote throughout, with concrete\nexamples and applications drawn from Judson. Each topic rests on the ones before\nit: everything is a set with structure, and the structure is the point.",{"title":15,"searchDepth":16,"depth":16,"links":17},"",2,[],[19,21,26,30,32,34,38,40,44,46,48,52,54],{"p":20},"Abstract algebra studies operations stripped to their axioms. A\n\u003Cstrong>group\u003C\u002Fstrong> is the leanest of them — a set with one\nassociative operation, an identity, and inverses — and this modest package\nalready captures the mathematics of \u003Cem>symmetry\u003C\u002Fem>.\n",{"fig":22,"n":23,"caption":24,"large":25},"aa-cyclic","001","A cyclic group: one generator sweeps out every element, ℤ\u002F6 as rotations\nof a hexagon.\n",true,{"fig":27,"n":28,"caption":29},"aa-dihedral","002","Dihedral symmetry: rotations and reflections of a triangle, the full group\nof its rigid motions.\n",{"p":31},"The first examples are the symmetries of a shape. Rotate a polygon and it\nlands on itself; reflect it and it does too. Compose those moves and you\nget a group — cyclic when only rotations count, \u003Cstrong>dihedral\u003C\u002Fstrong>\nonce reflections join in.\n",{"p":33},"Structure is enforced by \u003Cstrong>subgroups\u003C\u002Fstrong>. A subgroup sits inside\na group and splits it into \u003Cem>cosets\u003C\u002Fem> — translated copies, all the\nsame size — so the order of any subgroup must divide the order of the\nwhole. That single counting fact, Lagrange's theorem, constrains what\ngroups can even exist.\n",{"fig":35,"n":36,"caption":37},"aa-cosets","003","Cosets partition a group into equal blocks — the idea behind Lagrange's\ntheorem and quotient groups.\n",{"p":39},"To compare groups you use \u003Cstrong>homomorphisms\u003C\u002Fstrong>: maps that respect\nthe operation. Their reach is measured by the \u003Cem>image\u003C\u002Fem>, their\ncollapse by the \u003Cem>kernel\u003C\u002Fem>, and the two are tied together by the\nisomorphism theorems — a quotient by the kernel is a faithful copy of the\nimage.\n",{"fig":41,"n":42,"caption":43},"aa-homomorphism","004","A homomorphism ℤ\u002F6 → ℤ\u002F3: source elements collapse onto their images, the\nkernel mapping to the identity.\n",{"p":45},"Layer a second operation on and you get \u003Cstrong>rings\u003C\u002Fstrong> and\n\u003Cstrong>fields\u003C\u002Fstrong> — the axioms behind arithmetic. Polynomials,\nintegers, and matrices are all rings; fields are where division always\nworks, and their extensions are the setting for Galois theory.\n",{"p":47},"The finite structures reveal themselves in their \u003Cstrong>Cayley\ntables\u003C\u002Fstrong>. Every row and column is a permutation of the elements — a\nLatin square — which is just invertibility made visible.\n",{"fig":49,"n":50,"caption":51},"aa-cayley","005","A Cayley table: the whole operation written out, each row and column a\npermutation of the group.\n",{"p":53},"The payoff is \u003Cstrong>Galois theory\u003C\u002Fstrong>, which pins field extensions\nto groups of symmetries and settles classical questions — why the general\nquintic has no formula in radicals, why some angles cannot be trisected by\ncompass and straightedge.\n",{"p":55},"Learn the axioms once and the same skeleton appears everywhere: in number\ntheory, in geometry, in cryptography, in the symmetries of physics. Abstract\nalgebra is the grammar those subjects are written in.\n","math","Abstract algebra studies operations stripped to their axioms. Groups\ncapture symmetry; rings and fields capture arithmetic; modules generalize\nvector spaces; and Galois theory connects field extensions back to groups,\nsettling classical questions like the insolvability of the quintic. These\nnotes follow Dummit & Foote through groups, rings, modules, and fields,\nwith applications drawn from Judson.\n",false,"md",{},"\u002Fabstract-algebra",[],"---\ntitle: Abstract Algebra\nstatus: available\ncategory: math\nblurb: |\n  Groups, rings, and fields — the axioms behind symmetry and arithmetic, from\n  permutations and Sylow theorems to Galois theory and why the quintic has no\n  formula.\ndescription: |\n  Abstract algebra studies operations stripped to their axioms. Groups\n  capture symmetry; rings and fields capture arithmetic; modules generalize\n  vector spaces; and Galois theory connects field extensions back to groups,\n  settling classical questions like the insolvability of the quintic. These\n  notes follow Dummit & Foote through groups, rings, modules, and fields,\n  with applications drawn from Judson.\nbrief:\n  - p: |\n      Abstract algebra studies operations stripped to their axioms. A\n      \u003Cstrong>group\u003C\u002Fstrong> is the leanest of them — a set with one\n      associative operation, an identity, and inverses — and this modest package\n      already captures the mathematics of \u003Cem>symmetry\u003C\u002Fem>.\n  - fig: aa-cyclic\n    n: \"001\"\n    caption: |\n      A cyclic group: one generator sweeps out every element, ℤ\u002F6 as rotations\n      of a hexagon.\n    large: true\n  - fig: aa-dihedral\n    n: \"002\"\n    caption: |\n      Dihedral symmetry: rotations and reflections of a triangle, the full group\n      of its rigid motions.\n  - p: |\n      The first examples are the symmetries of a shape. Rotate a polygon and it\n      lands on itself; reflect it and it does too. Compose those moves and you\n      get a group — cyclic when only rotations count, \u003Cstrong>dihedral\u003C\u002Fstrong>\n      once reflections join in.\n  - p: |\n      Structure is enforced by \u003Cstrong>subgroups\u003C\u002Fstrong>. A subgroup sits inside\n      a group and splits it into \u003Cem>cosets\u003C\u002Fem> — translated copies, all the\n      same size — so the order of any subgroup must divide the order of the\n      whole. That single counting fact, Lagrange's theorem, constrains what\n      groups can even exist.\n  - fig: aa-cosets\n    n: \"003\"\n    caption: |\n      Cosets partition a group into equal blocks — the idea behind Lagrange's\n      theorem and quotient groups.\n  - p: |\n      To compare groups you use \u003Cstrong>homomorphisms\u003C\u002Fstrong>: maps that respect\n      the operation. Their reach is measured by the \u003Cem>image\u003C\u002Fem>, their\n      collapse by the \u003Cem>kernel\u003C\u002Fem>, and the two are tied together by the\n      isomorphism theorems — a quotient by the kernel is a faithful copy of the\n      image.\n  - fig: aa-homomorphism\n    n: \"004\"\n    caption: |\n      A homomorphism ℤ\u002F6 → ℤ\u002F3: source elements collapse onto their images, the\n      kernel mapping to the identity.\n  - p: |\n      Layer a second operation on and you get \u003Cstrong>rings\u003C\u002Fstrong> and\n      \u003Cstrong>fields\u003C\u002Fstrong> — the axioms behind arithmetic. Polynomials,\n      integers, and matrices are all rings; fields are where division always\n      works, and their extensions are the setting for Galois theory.\n  - p: |\n      The finite structures reveal themselves in their \u003Cstrong>Cayley\n      tables\u003C\u002Fstrong>. Every row and column is a permutation of the elements — a\n      Latin square — which is just invertibility made visible.\n  - fig: aa-cayley\n    n: \"005\"\n    caption: |\n      A Cayley table: the whole operation written out, each row and column a\n      permutation of the group.\n  - p: |\n      The payoff is \u003Cstrong>Galois theory\u003C\u002Fstrong>, which pins field extensions\n      to groups of symmetries and settles classical questions — why the general\n      quintic has no formula in radicals, why some angles cannot be trisected by\n      compass and straightedge.\n  - p: |\n      Learn the axioms once and the same skeleton appears everywhere: in number\n      theory, in geometry, in cryptography, in the symmetries of physics. Abstract\n      algebra is the grammar those subjects are written in.\n---\n\nTwo ideas run through the whole subject: an operation, and the symmetry it\nencodes. The sequence starts with groups — their axioms, the cyclic and\ndihedral examples, permutations and the symmetric group — then develops the\nmachinery that makes them tractable: subgroups and cosets, Lagrange's theorem,\nnormal subgroups and quotients, homomorphisms and the isomorphism theorems, and\nthe Sylow theorems that dissect finite groups by prime. From there it adds a\nsecond operation to reach rings (ideals, quotients, polynomial and Euclidean\ndomains, unique factorization) and modules (the structure theorem that classifies\nthem over a PID, with linear algebra and finitely generated abelian groups as\ncorollaries). The last movement is field theory and Galois theory, connecting\nfield extensions back to groups and resolving the classical impossibilities —\nthe insolvability of the quintic, the constructions that compass and straightedge\ncannot reach. These notes follow Dummit & Foote throughout, with concrete\nexamples and applications drawn from Judson. Each topic rests on the ones before\nit: everything is a set with structure, and the structure is the point.\n\n",{"text":65,"minutes":66,"time":67,"words":68},"1 min 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Functions, and Equivalence Relations","\u002Fabstract-algebra\u002Ffoundations\u002Fsets-functions-relations",[989],"Algebra is built on three prior notions: the set, the map between sets, and the equivalence relation that reorganizes a set into disjoint classes. Sets, maps (injective, surjective, bijective), fibers and preimages, and the correspondence between equivalence relations and partitions — the one structural fact reused in every later quotient construction.\n",{"title":999,"path":1000,"lessonNumber":16,"topics":1001,"summary":1002},"The Integers and Modular Arithmetic","\u002Fabstract-algebra\u002Ffoundations\u002Fintegers-and-modular-arithmetic",[989],"The integers carry the template every ring later imitates: well-ordering drives induction, induction drives the division algorithm, and division drives the Euclidean algorithm, gcd, Bézout's identity, and unique factorization into primes. Quotienting by congruence mod n builds the first finite arithmetic, Z\u002FnZ, whose invertible elements form the group of units.\n",{"module":1004,"moduleNumber":16,"slug":1005,"lessons":1006},"Groups and Symmetry","groups-and-symmetry",[1007,1012,1017,1023],{"title":1008,"path":1009,"lessonNumber":990,"topics":1010,"summary":1011},"Group Axioms and First Examples","\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fgroup-axioms-and-first-examples",[1004],"A group is a set with one associative operation that has an identity and inverses. We state the axioms, prove that the identity, inverses, and cancellation behave as expected, define the order of a group and of an element, and catalogue the running examples: the integers, the additive group of residues mod n, and the multiplicative group of units mod n.\n",{"title":1013,"path":1014,"lessonNumber":16,"topics":1015,"summary":1016},"Dihedral and Symmetric Groups","\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fdihedral-and-symmetric-groups",[1004],"The dihedral group D_{2n} is the symmetries of a regular n-gon, generated by a rotation r and a reflection s subject to three relations. The symmetric group S_n is all permutations of n objects, written in cycle notation. Orders, generators and relations, cycle decomposition, the order of a permutation from its cycle type, and the parity that splits S_n in half.\n",{"title":1018,"path":1019,"lessonNumber":1020,"topics":1021,"summary":1022},"Matrix and Quaternion Groups","\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fmatrix-and-quaternion-groups",3,[1004],"Invertible matrices over a field form the general linear group GL_n(F), with the determinant-one matrices as the subgroup SL_n(F). Over a finite field the order of GL_n(F) has a clean product formula. The quaternion group Q_8 is a second small nonabelian group, distinct from the dihedral group of the same order; its multiplication and subgroup structure sharpen the contrast between the two.\n",{"title":1024,"path":1025,"lessonNumber":1026,"topics":1027,"summary":1028},"Homomorphisms, Isomorphisms, and Actions","\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fhomomorphisms-and-group-actions",4,[1004],"A homomorphism is a map between groups that respects the operation; an isomorphism is a bijective one, making two groups the same up to relabeling. The kernel and image measure how far a homomorphism is from injective and surjective. A group action realizes a group as permutations of a set, and actions correspond exactly to homomorphisms into a symmetric group, with orbits and stabilizers as the first tools for counting.\n",{"module":1030,"moduleNumber":1020,"slug":1031,"lessons":1032},"Subgroups and Quotients","subgroups-and-quotients",[1033,1038,1043,1048,1053,1059],{"title":1034,"path":1035,"lessonNumber":990,"topics":1036,"summary":1037},"Subgroups and Their Substructures","\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fsubgroups-and-substructures",[1030],"A subgroup is a subset that is a group under the inherited operation. One test decides it: nonempty and closed under the map $(x,y) \\mapsto xy^{-1}$. From an arbitrary subset $A$ we build the centralizer, normalizer, and center, and from an action the stabilizer and kernel, all of them subgroups nested in a fixed chain inside $G$.\n",{"title":1039,"path":1040,"lessonNumber":16,"topics":1041,"summary":1042},"Cyclic Groups","\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcyclic-groups",[1030],"A cyclic group is generated by one element. Two facts organize the whole theory: the order of an element equals the order of the subgroup it generates, and cyclic groups of equal order are isomorphic, so $\\mathbb{Z}$ and $\\mathbb{Z}\u002Fn\\mathbb{Z}$ are the only ones. From there the generators ($\\varphi(n)$ of them), the subgroups (one per divisor of $n$), and a fast exponentiation algorithm all follow.\n",{"title":1044,"path":1045,"lessonNumber":1020,"topics":1046,"summary":1047},"Generation and the Lattice of Subgroups","\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fgeneration-and-subgroup-lattices",[1030],"The subgroup generated by a subset $A$ is the smallest subgroup containing it, described top-down as an intersection and bottom-up as the set of words in $A$ and its inverses. Collecting all subgroups and ordering them by containment produces the subgroup lattice, whose Hasse diagram shows the joins, meets, and containment relations among all subgroups.\n",{"title":1049,"path":1050,"lessonNumber":1026,"topics":1051,"summary":1052},"Cosets, Lagrange, and Normal Subgroups","\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcosets-lagrange-and-normal-subgroups",[1030],"The left cosets of a subgroup partition a group into equal-sized blocks, so the order of a subgroup divides the order of the group: Lagrange's theorem. When the blocks can be multiplied consistently — exactly when the subgroup is normal — they form the quotient group $G\u002FN$. Fermat's and Euler's theorems fall out as index computations.\n",{"title":1054,"path":1055,"lessonNumber":1056,"topics":1057,"summary":1058},"The Isomorphism Theorems","\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fisomorphism-theorems",5,[1030],"Four theorems relate homomorphisms, quotients, and subgroup lattices. The first identifies the image of a homomorphism with the quotient by its kernel; the second and third compute quotients built from two subgroups and quotients of quotients; the fourth matches the subgroups of $G\u002FN$ with the subgroups of $G$ lying above $N$. Together they make quotient groups computable.\n",{"title":1060,"path":1061,"lessonNumber":1062,"topics":1063,"summary":1064},"Composition Series and the Alternating Group","\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcomposition-series-and-the-alternating-group",6,[1030],"A composition series breaks a finite group into simple quotient factors, and Jordan-Hölder says those factors are unique up to order. This turns classification into two problems: list the simple groups, and describe how to reassemble them. The sign homomorphism splits $S_n$ into even and odd permutations, defining the alternating group $A_n$, simple for $n \\ge 5$.\n",{"module":1066,"moduleNumber":1026,"slug":1067,"lessons":1068},"Group Actions and Sylow Theory","group-actions-and-sylow",[1069,1074,1079,1084],{"title":1070,"path":1071,"lessonNumber":990,"topics":1072,"summary":1073},"Actions, Orbits, and Cayley's Theorem","\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Factions-and-cayleys-theorem",[1066],"A group action turns abstract elements into permutations of a set. The action splits the set into orbits, and the orbit-stabilizer theorem ties each orbit's size to the index of a stabilizer. Applied to a group acting on itself by left multiplication, this gives Cayley's theorem: every group is a group of permutations.\n",{"title":1075,"path":1076,"lessonNumber":16,"topics":1077,"summary":1078},"Conjugation and the Class Equation","\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fconjugation-and-the-class-equation",[1066],"A group acts on itself by conjugation, and the orbits are the conjugacy classes. Orbit-stabilizer turns the resulting partition into the class equation, which forces every group of prime-power order to have a nontrivial center. Conjugacy in the symmetric group is cycle type, and Burnside's lemma counts orbits by averaging fixed points.\n",{"title":1080,"path":1081,"lessonNumber":1020,"topics":1082,"summary":1083},"The Sylow Theorems","\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fsylow-theorems",[1066],"Lagrange's theorem forbids subgroups whose order fails to divide the group order; Sylow's theorems supply a partial converse for prime powers. A Sylow p-subgroup always exists, all of them are conjugate, and their count satisfies two congruence-and-divisibility constraints tight enough to prove many groups non-simple from their order alone.\n",{"title":1085,"path":1086,"lessonNumber":1026,"topics":1087,"summary":1088},"Automorphisms and Simplicity of Aₙ","\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fautomorphisms-and-simple-groups",[1066],"Conjugation makes a group act on itself and on its normal subgroups by automorphisms, giving the inner automorphism group G\u002FZ(G) and the embedding of N(H)\u002FC(H) into Aut(H). Characteristic subgroups are those every automorphism fixes, and the automorphism group of a cyclic group is its unit group. The lesson closes by proving the alternating group Aₙ is simple for n ≥ 5.\n",{"module":1090,"moduleNumber":1056,"slug":1091,"lessons":1092},"Products and Group Structure","products-and-group-structure",[1093,1098,1103,1108],{"title":1094,"path":1095,"lessonNumber":990,"topics":1096,"summary":1097},"Direct Products and Finite Abelian Groups","\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fdirect-products-and-finite-abelian-groups",[1090],"The direct product assembles a larger group from componentwise copies of smaller ones, and a recognition theorem reverses the process when two normal subgroups meet trivially and span the group. The Fundamental Theorem of Finitely Generated Abelian Groups then classifies every such group by two equivalent invariants, invariant factors and elementary divisors.\n",{"title":1099,"path":1100,"lessonNumber":16,"topics":1101,"summary":1102},"Semidirect Products","\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fsemidirect-products",[1090],"The semidirect product relaxes the direct product by requiring only one factor to be normal, with the other acting on it through a homomorphism into its automorphism group. This single twisting map lets abelian pieces assemble into non-abelian groups, realizes the dihedral groups as $\\mathbb{Z}_n \\rtimes \\mathbb{Z}_2$, and, with a recognition theorem, classifies groups of several small orders.\n",{"title":1104,"path":1105,"lessonNumber":1020,"topics":1106,"summary":1107},"p-Groups, Nilpotent, and Solvable Groups","\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fnilpotent-and-solvable-groups",[1090],"Finite p-groups have nontrivial center, and iterating the center upward builds the nilpotent groups, which decompose as the direct product of their Sylow subgroups. Iterating the commutator downward builds the solvable groups, whose factors are abelian. The chain cyclic, abelian, nilpotent, solvable orders these classes, and A_5 breaks the last link.\n",{"title":1109,"path":1110,"lessonNumber":1026,"topics":1111,"summary":1112},"Classifying Groups of Small Order","\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fclassifying-small-groups",[1090],"With Sylow's theorem to force normal subgroups, direct and semidirect products to assemble them, and presentations to name the result, every group up to order fifteen can be listed explicitly. Free groups make presentations precise: generators with no relations, from which any group is a quotient by the normal closure of its relations.\n",{"module":1114,"moduleNumber":1062,"slug":1115,"lessons":1116},"Ring Theory","ring-theory",[1117,1122,1127],{"title":1118,"path":1119,"lessonNumber":990,"topics":1120,"summary":1121},"Rings: Definitions and Examples","\u002Fabstract-algebra\u002Fring-theory\u002Frings-definitions-and-examples",[1114],"A ring carries two operations: an abelian group under addition and an associative multiplication linked by the distributive laws. The named special cases — commutative rings, integral domains, division rings, and fields — differ only in how their multiplication behaves. Standard examples include quadratic integer rings, polynomial rings, matrix rings, and group rings.\n",{"title":1123,"path":1124,"lessonNumber":16,"topics":1125,"summary":1126},"Ideals, Quotient Rings, and Homomorphisms","\u002Fabstract-algebra\u002Fring-theory\u002Fideals-quotients-and-homomorphisms",[1114],"Ring homomorphisms have kernels that absorb multiplication; such subsets are ideals, and every ideal is the kernel of the projection onto a quotient ring. The quotient construction yields the ring isomorphism theorems and classifies ideals by their quotients: R\u002FI is a field exactly when I is maximal, an integral domain exactly when I is prime.\n",{"title":1128,"path":1129,"lessonNumber":1020,"topics":1130,"summary":1131},"Fields of Fractions and the CRT","\u002Fabstract-algebra\u002Fring-theory\u002Ffractions-and-the-chinese-remainder-theorem",[1114],"Rings of fractions invert a multiplicatively closed set, enlarging an integral domain into its field of fractions the way Z becomes Q. The Chinese Remainder Theorem splits a quotient by comaximal ideals into a direct product, generalizing Z\u002FmnZ ≅ Z\u002FmZ × Z\u002FnZ and explaining why the Euler function is multiplicative.\n",{"module":1133,"moduleNumber":1134,"slug":1135,"lessons":1136},"Factorization and Polynomial Rings",7,"factorization-and-polynomials",[1137,1142,1147,1152],{"title":1138,"path":1139,"lessonNumber":990,"topics":1140,"summary":1141},"Euclidean Domains, PIDs, and UFDs","\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Feuclidean-domains-pids-ufds",[1133],"Three classes of integral domain, ordered by how much of elementary arithmetic survives: Euclidean domains carry a division algorithm, principal ideal domains make every ideal a single multiple, and unique factorization domains factor every element into irreducibles in one way. We prove the chain ED implies PID implies UFD, the classes are separated by explicit counterexamples, and irreducible and prime coincide exactly in a UFD.\n",{"title":1143,"path":1144,"lessonNumber":16,"topics":1145,"summary":1146},"Polynomial Rings over Fields","\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fpolynomial-rings-over-fields",[1133],"When the coefficients form a field, polynomial long division works exactly as it does over the rationals, and it works with a unique quotient and remainder. That single fact makes F[x] a Euclidean domain, hence a PID and a UFD: every ideal is the multiples of one polynomial, roots correspond to linear factors, and F[x]\u002F(f) is a field precisely when f is irreducible.\n",{"title":1148,"path":1149,"lessonNumber":1020,"topics":1150,"summary":1151},"Gauss's Lemma and Unique Factorization","\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fgauss-lemma-and-unique-factorization",[1133],"A UFD is not a field, so its polynomial ring is not a PID — yet unique factorization survives the passage from R to R[x]. Gauss's lemma supplies the passage: a polynomial that factors over the fraction field already factors over R, once content is factored out. This gives the theorem that R[x] is a UFD whenever R is, so Z[x] and Q[x,y] factor uniquely even though neither is a PID.\n",{"title":1153,"path":1154,"lessonNumber":1026,"topics":1155,"summary":1156},"Irreducibility Criteria and Gröbner Bases","\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Firreducibility-criteria-and-groebner",[1133],"Deciding whether a given polynomial is irreducible, and computing in multivariate polynomial rings. In one variable: the rational root test, reduction modulo a prime, and Eisenstein's criterion. In several variables, where division fails, a monomial order gives leading terms, a Gröbner basis restores a well-defined remainder, and Buchberger's algorithm computes it.\n",{"module":1158,"moduleNumber":1159,"slug":1160,"lessons":1161},"Module Theory",8,"module-theory",[1162,1167,1172,1177],{"title":1163,"path":1164,"lessonNumber":990,"topics":1165,"summary":1166},"Introduction to Modules","\u002Fabstract-algebra\u002Fmodule-theory\u002Fintroduction-to-modules",[1158],"A module is an abelian group on which a ring acts, generalizing both vector spaces (when the ring is a field) and abelian groups (when the ring is the integers). Submodules, homomorphisms, quotients, and the isomorphism theorems carry over from groups, and an F[x]-module is the same datum as a vector space with a chosen linear operator — the correspondence behind the canonical forms.\n",{"title":1168,"path":1169,"lessonNumber":16,"topics":1170,"summary":1171},"Generation, Direct Sums, and Free Modules","\u002Fabstract-algebra\u002Fmodule-theory\u002Ffree-modules-and-direct-sums",[1158],"A generating set spans a module by R-linear combinations; a direct sum decomposes it into independent pieces; a free module has a basis and the universal property that a homomorphism is determined by arbitrary values on that basis. Rank is well defined over a commutative ring, torsion blocks a basis, and every module is a quotient of a free one — a presentation by generators and relations.\n",{"title":1173,"path":1174,"lessonNumber":1020,"topics":1175,"summary":1176},"Tensor Products and Exact Sequences","\u002Fabstract-algebra\u002Fmodule-theory\u002Ftensor-products-and-exact-sequences",[1158],"The tensor product builds a module in which elements of two modules can be multiplied, characterized by a universal property turning bilinear maps into linear ones; extension of scalars is its guiding case. Exact sequences track how a module is assembled from a submodule and a quotient, when that assembly splits, and which modules — projective, injective, flat — make the Hom and tensor functors preserve exactness.\n",{"title":1178,"path":1179,"lessonNumber":1026,"topics":1180,"summary":1181},"Vector Spaces and Linear Maps","\u002Fabstract-algebra\u002Fmodule-theory\u002Fvector-spaces-and-linear-maps",[1158],"A vector space is a module over a field, and the field hypothesis removes every pathology a general module can have: every vector space is free, so it has a basis, a well-defined dimension, and a coordinate isomorphism with F^n. Linear maps become matrices, change of basis becomes similarity, every space pairs with a dual of the same dimension, and the determinant is the unique alternating multilinear normalized form.\n",{"module":1183,"moduleNumber":1184,"slug":1185,"lessons":1186},"Modules over PIDs and Canonical Forms",9,"modules-over-pids",[1187,1192,1197],{"title":1188,"path":1189,"lessonNumber":990,"topics":1190,"summary":1191},"The Structure Theorem for Modules over a PID","\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fstructure-theorem-over-pids",[1183],"Every finitely generated module over a principal ideal domain splits as a free part plus a direct sum of cyclic torsion pieces, in two canonical ways: invariant factors, tied together by a divisibility chain, and elementary divisors, one prime power at a time. Existence follows from the stacked-basis theorem, both lists are unique, and the case $R = \\mathbb{Z}$ is the classification of finitely generated abelian groups.\n",{"title":1193,"path":1194,"lessonNumber":16,"topics":1195,"summary":1196},"Rational Canonical Form","\u002Fabstract-algebra\u002Fmodules-over-pids\u002Frational-canonical-form",[1183],"A linear operator turns its vector space into a module over the polynomial ring $F[x]$, with $x$ acting as the operator. The structure theorem's invariant factors then become polynomials, each cyclic summand becomes a companion matrix, and the block-diagonal assembly is the rational canonical form. It is unique, it is computed inside the base field, and two matrices are similar exactly when their rational canonical forms agree.\n",{"title":1198,"path":1199,"lessonNumber":1020,"topics":1200,"summary":1201},"Jordan Canonical Form","\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fjordan-canonical-form",[1183],"When the base field contains all the eigenvalues, the elementary divisors of an operator are powers of linear polynomials, and each cyclic summand becomes a Jordan block: an eigenvalue on the diagonal with ones just above it. Stacking the blocks gives the Jordan canonical form, unique up to reordering, as close to diagonal as the operator allows. Diagonalizability reads off the minimal polynomial, and the block sizes are counted by ranks of powers of the operator minus the eigenvalue.\n",{"module":1203,"moduleNumber":1204,"slug":1205,"lessons":1206},"Field Theory",10,"field-theory",[1207,1212,1217,1222],{"title":1208,"path":1209,"lessonNumber":990,"topics":1210,"summary":1211},"Field Extensions and Algebraic Elements","\u002Fabstract-algebra\u002Ffield-theory\u002Ffield-extensions-and-algebraic-elements",[1203],"A field extension makes a larger field K into a vector space over a smaller field F, and its degree [K:F] is that dimension. Adjoining a root of an irreducible polynomial builds a simple extension F(α) isomorphic to F[x]\u002F(m), whose degree is the degree of the minimal polynomial. The tower law makes these degrees multiply, which turns algebra over fields into bookkeeping with integers.\n",{"title":1213,"path":1214,"lessonNumber":16,"topics":1215,"summary":1216},"Straightedge-and-Compass Constructions","\u002Fabstract-algebra\u002Ffield-theory\u002Fstraightedge-and-compass-constructions",[1203],"The lengths a straightedge and compass can build from a unit form a field closed under square roots, and every constructible number lies in a tower of quadratic extensions. So its degree over the rationals is a power of two. That single obstruction settles three problems the Greeks left open: doubling the cube, trisecting a general angle, and squaring the circle are all impossible.\n",{"title":1218,"path":1219,"lessonNumber":1020,"topics":1220,"summary":1221},"Splitting Fields and Algebraic Closure","\u002Fabstract-algebra\u002Ffield-theory\u002Fsplitting-fields-and-algebraic-closure",[1203],"The splitting field of a polynomial is the smallest extension in which it factors into linear pieces, obtained by adjoining all its roots. Every polynomial has one, its degree is at most n factorial, and any two splitting fields are isomorphic. Pushing this to all polynomials at once gives the algebraic closure, a field in which every polynomial splits and which is unique up to isomorphism.\n",{"title":1223,"path":1224,"lessonNumber":1026,"topics":1225,"summary":1226},"Separable Extensions and Cyclotomic Fields","\u002Fabstract-algebra\u002Ffield-theory\u002Fseparable-and-cyclotomic-extensions",[1203],"A polynomial is separable when its roots are distinct, detected by whether it shares a factor with its formal derivative. Over perfect fields — characteristic zero and finite fields — every irreducible is separable, and the existence and uniqueness of the finite fields follow. Cyclotomic polynomials package the roots of unity by order, are irreducible over the rationals, and give the cyclotomic field its degree phi(n).\n",{"module":1228,"moduleNumber":1229,"slug":1230,"lessons":1231},"Galois Theory",11,"galois-theory",[1232,1237,1242,1247,1252],{"title":1233,"path":1234,"lessonNumber":990,"topics":1235,"summary":1236},"The Galois Correspondence","\u002Fabstract-algebra\u002Fgalois-theory\u002Fthe-galois-correspondence",[1228],"Galois theory attaches to a field extension its group of symmetries and shows that, for the right extensions, the subgroups of that group are in exact order-reversing correspondence with the intermediate fields. The automorphism group, Artin's theorem, the characterization of Galois extensions, and the Fundamental Theorem together turn questions about fields into questions about finite groups.\n",{"title":1238,"path":1239,"lessonNumber":16,"topics":1240,"summary":1241},"Finite Fields","\u002Fabstract-algebra\u002Fgalois-theory\u002Ffinite-fields",[1228],"Every finite field has prime-power order, is the splitting field of $x^{p^n} - x$, and is unique up to isomorphism. Its extension over the prime field is Galois with cyclic group generated by the Frobenius map $x \\mapsto x^p$, so the Galois correspondence reduces the subfield lattice to the divisor lattice of $n$. Möbius inversion counts the irreducible polynomials of each degree, and cyclic error-correcting codes are one application.\n",{"title":1243,"path":1244,"lessonNumber":1020,"topics":1245,"summary":1246},"Cyclotomic and Abelian Extensions","\u002Fabstract-algebra\u002Fgalois-theory\u002Fcyclotomic-and-abelian-extensions",[1228],"The Galois group of the $n$th cyclotomic field over $\\mathbb{Q}$ is the unit group $(\\mathbb{Z}\u002Fn\\mathbb{Z})^\\times$, which makes cyclotomic fields the worked catalogue of abelian extensions of $\\mathbb{Q}$. The isomorphism identifies subfields with subgroups, realizes every finite abelian group as a Galois group over $\\mathbb{Q}$, and leads to Kronecker–Weber. Composites of Galois extensions and the primitive element theorem supply the machinery.\n",{"title":1248,"path":1249,"lessonNumber":1026,"topics":1250,"summary":1251},"Galois Groups of Polynomials","\u002Fabstract-algebra\u002Fgalois-theory\u002Fgalois-groups-of-polynomials",[1228],"Ordering the roots of a separable polynomial embeds its Galois group in the symmetric group $S_n$, and the group is transitive exactly when the polynomial is irreducible. The discriminant decides membership in $A_n$; for cubics and quartics the resolvent cubic pins the group down; and reduction modulo a prime produces elements of prescribed cycle type, the standard tool for computing Galois groups over $\\mathbb{Q}$.\n",{"title":1253,"path":1254,"lessonNumber":1056,"topics":1255,"summary":1256},"Solvability by Radicals and the Quintic","\u002Fabstract-algebra\u002Fgalois-theory\u002Fsolvability-by-radicals-and-the-quintic",[1228],"A polynomial is solvable by radicals exactly when its Galois group is solvable. Cyclic extensions are radical extensions once roots of unity are present, which turns a radical tower into a solvable subnormal series. Since $S_n$ is solvable only for $n \\le 4$, the general quintic has no radical formula, and an explicit quintic with Galois group $S_5$ has roots provably not expressible in radicals.\n",{"module":1258,"moduleNumber":1259,"slug":1260,"lessons":1261},"Capstone: Where Algebra Goes Next",12,"capstone",[1262,1267],{"title":1263,"path":1264,"lessonNumber":990,"topics":1265,"summary":1266},"A Glimpse of Commutative Algebra and Algebraic Geometry","\u002Fabstract-algebra\u002Fcapstone\u002Fcommutative-algebra-and-algebraic-geometry",[1258],"Commutative algebra reads geometry off the ring of polynomial functions. The dictionary runs through Noetherian rings and the ascending chain condition, Hilbert's Basis Theorem, affine algebraic sets and the two maps connecting ideals to zero sets, radicals, the Zariski topology, and Hilbert's Nullstellensatz, which over an algebraically closed field makes radical ideals and algebraic sets the same object.\n",{"title":1268,"path":1269,"lessonNumber":16,"topics":1270,"summary":1271},"A Glimpse of Representation and Character Theory","\u002Fabstract-algebra\u002Fcapstone\u002Frepresentation-and-character-theory",[1258],"Representation theory studies a group by the ways it can act linearly on a vector space. Representations are equivalent to modules over the group ring; Maschke's theorem gives complete reducibility, the Wedderburn consequences bound the irreducible degrees, and character theory reduces a representation to a trace invariant governed by the orthogonality relations and displayed in the character table of a small group.\n",1786059478799]