Abstract algebra

Inmathematics,more specificallyalgebra,abstract algebraormodern algebrais the study ofalgebraic structures,which aresetswith specificoperationsacting on their elements.^[1]Algebraic structures includegroups,rings,fields,modules,vector spaces,lattices,andalgebras over a field.The termabstract algebrawas coined in the early 20th century to distinguish it from older parts of algebra, and more specifically fromelementary algebra,the use ofvariablesto represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except inpedagogy.

Algebraic structures, with their associatedhomomorphisms,formmathematical categories.Category theorygives a unified framework to study properties and constructions that are similar for various structures.

Universal algebrais a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called thevarietyof groups.

History[edit]

Before the nineteenth century,algebrawas defined as the study ofpolynomials.^[2]Abstract algebra came into existence during the nineteenth century as more complex problems and solution methods developed. Concrete problems and examples came from number theory, geometry, analysis, and the solutions ofalgebraic equations.Most theories that are now recognized as parts of abstract algebra started as collections of disparate facts from various branches of mathematics, acquired a common theme that served as a core around which various results were grouped, and finally became unified on a basis of a common set of concepts. This unification occurred in the early decades of the 20th century and resulted in the formalaxiomaticdefinitions of variousalgebraic structuressuch as groups, rings, and fields.^[3]This historical development is almost the opposite of the treatment found in popular textbooks, such as van der Waerden'sModerne Algebra,^[4]which start each chapter with a formal definition of a structure and then follow it with concrete examples.^[5]

Elementary algebra[edit]

The study of polynomial equations oralgebraic equationshas a long history.c. 1700 BC,the Babylonians were able to solve quadratic equations specified as word problems. This word problem stage is classified asrhetorical algebraand was the dominant approach up to the 16th century.Al-Khwarizmioriginated the word "algebra" in 830 AD, but his work was entirely rhetorical algebra. Fully symbolic algebra did not appear untilFrançois Viète's 1591New Algebra,and even this had some spelled out words that were given symbols in Descartes's 1637La Géométrie.^[6]The formal study of solving symbolic equations ledLeonhard Eulerto accept what were then considered "nonsense" roots such asnegative numbersandimaginary numbers,in the late 18th century.^[7]However, European mathematicians, for the most part, resisted these concepts until the middle of the 19th century.^[8]

George Peacock's 1830Treatise of Algebrawas the first attempt to place algebra on a strictly symbolic basis. He distinguished a newsymbolical algebra,distinct from the oldarithmetical algebra.Whereas in arithmetical algebra $a-b$ is restricted to $a\geq b$ ,in symbolical algebra all rules of operations hold with no restrictions. Using this Peacock could show laws such as $(-a)(-b)=ab$ ,by letting $a=0,c=0$ in $(a-b)(c-d)=ac+bd-ad-bc$ .Peacock used what he termed theprinciple of the permanence of equivalent formsto justify his argument, but his reasoning suffered from theproblem of induction.^[9]For example, ${\sqrt {a}}{\sqrt {b}}={\sqrt {ab}}$ holds for the nonnegativereal numbers,but not for generalcomplex numbers.

Early group theory[edit]

Several areas of mathematics led to the study of groups. Lagrange's 1770 study of the solutions of the quintic equation led to theGalois group of a polynomial.Gauss's 1801 study ofFermat's little theoremled to thering of integers modulo n,themultiplicative group of integers modulo n,and the more general concepts ofcyclic groupsandabelian groups.Klein's 1872Erlangen programstudied geometry and led tosymmetry groupssuch as theEuclidean groupand the group ofprojective transformations.In 1874 Lie introduced the theory ofLie groups,aiming for "the Galois theory of differential equations". In 1876 Poincaré and Klein introduced the group ofMöbius transformations,and its subgroups such as themodular groupandFuchsian group,based on work on automorphic functions in analysis.^[10]

The abstract concept of group emerged slowly over the middle of the nineteenth century. Galois in 1832 was the first to use the term "group",^[11]signifying a collection of permutations closed under composition.^[12]Arthur Cayley's 1854 paperOn the theory of groupsdefined a group as a set with an associative composition operation and the identity 1, today called amonoid.^[13]In 1870 Kronecker defined an abstract binary operation that was closed, commutative, associative, and had the leftcancellation property $b\neq c\to a\cdot b\neq a\cdot c$ ,^[14]similar to the modern laws for a finiteabelian group.^[15]Weber's 1882 definition of a group was a closed binary operation that was associative and had left and right cancellation.^[16]Walther von Dyckin 1882 was the first to require inverse elements as part of the definition of a group.^[17]

Once this abstract group concept emerged, results were reformulated in this abstract setting. For example,Sylow's theoremwas reproven by Frobenius in 1887 directly from the laws of a finite group, although Frobenius remarked that the theorem followed from Cauchy's theorem on permutation groups and the fact that every finite group is a subgroup of a permutation group.^[18]^[19]Otto Hölderwas particularly prolific in this area, defining quotient groups in 1889, group automorphisms in 1893, as well as simple groups. He also completed theJordan–Hölder theorem.Dedekind and Miller independently characterizedHamiltonian groupsand introduced the notion of thecommutatorof two elements. Burnside, Frobenius, and Molien created therepresentation theoryof finite groups at the end of the nineteenth century.^[18]J. A. de Séguier's 1905 monographElements of the Theory of Abstract Groupspresented many of these results in an abstract, general form, relegating "concrete" groups to an appendix, although it was limited to finite groups. The first monograph on both finite and infinite abstract groups was O. K. Schmidt's 1916Abstract Theory of Groups.^[20]

Early ring theory[edit]

Noncommutative ring theory began with extensions of the complex numbers tohypercomplex numbers,specificallyWilliam Rowan Hamilton'squaternionsin 1843. Many other number systems followed shortly. In 1844, Hamilton presentedbiquaternions,Cayley introducedoctonions,and Grassman introducedexterior algebras.^[21]James Cocklepresentedtessarinesin 1848^[22]andcoquaternionsin 1849.^[23]William Kingdon Cliffordintroducedsplit-biquaternionsin 1873. In addition Cayley introducedgroup algebrasover the real and complex numbers in 1854 andsquare matricesin two papers of 1855 and 1858.^[24]

Once there were sufficient examples, it remained to classify them. In an 1870 monograph,Benjamin Peirceclassified the more than 150 hypercomplex number systems of dimension below 6, and gave an explicit definition of anassociative algebra.He defined nilpotent and idempotent elements and proved that any algebra contains one or the other. He also defined thePeirce decomposition.Frobenius in 1878 andCharles Sanders Peircein 1881 independently proved that the only finite-dimensional division algebras over $\mathbb {R}$ were the real numbers, the complex numbers, and the quaternions. In the 1880s Killing and Cartan showed that semisimpleLie algebrascould be decomposed into simple ones, and classified all simple Lie algebras. Inspired by this, in the 1890s Cartan, Frobenius, and Molien proved (independently) that a finite-dimensional associative algebra over $\mathbb {R}$ or $\mathbb {C}$ uniquely decomposes into thedirect sumsof a nilpotent algebra and a semisimple algebra that is the product of some number ofsimple algebras,square matrices over division algebras. Cartan was the first to define concepts such as direct sum and simple algebra, and these concepts proved quite influential. In 1907 Wedderburn extended Cartan's results to an arbitrary field, in what are now called theWedderburn principal theoremandArtin–Wedderburn theorem.^[25]

For commutative rings, several areas together led to commutative ring theory.^[26]In two papers in 1828 and 1832, Gauss formulated theGaussian integersand showed that they form aunique factorization domain(UFD) and proved thebiquadratic reciprocitylaw. Jacobi and Eisenstein at around the same time proved acubic reciprocitylaw for theEisenstein integers.^[25]The study ofFermat's last theoremled to thealgebraic integers.In 1847,Gabriel Laméthought he had proven FLT, but his proof was faulty as he assumed all thecyclotomic fieldswere UFDs, yet as Kummer pointed out, $\mathbb {Q} (\zeta _{23}))$ was not a UFD.^[27]In 1846 and 1847 Kummer introducedideal numbersand proved unique factorization into ideal primes for cyclotomic fields.^[28]Dedekind extended this in 1871 to show that every nonzero ideal in the domain of integers of an algebraic number field is a unique product ofprime ideals,a precursor of the theory ofDedekind domains.Overall, Dedekind's work created the subject ofalgebraic number theory.^[29]

In the 1850s, Riemann introduced the fundamental concept of aRiemann surface.Riemann's methods relied on an assumption he calledDirichlet's principle,^[30]which in 1870 was questioned by Weierstrass. Much later, in 1900, Hilbert justified Riemann's approach by developing thedirect method in the calculus of variations.^[31]In the 1860s and 1870s, Clebsch, Gordan, Brill, and especiallyM. Noetherstudiedalgebraic functionsand curves. In particular, Noether studied what conditions were required for a polynomial to be an element of the ideal generated by two algebraic curves in the polynomial ring $\mathbb {R} [x,y]$ ,although Noether did not use this modern language. In 1882 Dedekind and Weber, in analogy with Dedekind's earlier work on algebraic number theory, created a theory ofalgebraic function fieldswhich allowed the first rigorous definition of a Riemann surface and a rigorous proof of theRiemann–Roch theorem.Kronecker in the 1880s, Hilbert in 1890, Lasker in 1905, and Macauley in 1913 further investigated the ideals of polynomial rings implicit inE. Noether's work. Lasker proved a special case of theLasker-Noether theorem,namely that every ideal in a polynomial ring is a finite intersection ofprimary ideals.Macauley proved the uniqueness of this decomposition.^[32]Overall, this work led to the development ofalgebraic geometry.^[26]

In 1801 Gauss introducedbinary quadratic formsover the integers and defined theirequivalence.He further defined thediscriminantof these forms, which is aninvariant of a binary form.Between the 1860s and 1890sinvariant theorydeveloped and became a major field of algebra. Cayley, Sylvester, Gordan and others found theJacobianand theHessianfor binary quartic forms and cubic forms.^[33]In 1868 Gordan proved that thegraded algebraof invariants of a binary form over the complex numbers was finitely generated, i.e., has a basis.^[34]Hilbert wrote a thesis on invariants in 1885 and in 1890 showed that any form of any degree or number of variables has a basis. He extended this further in 1890 toHilbert's basis theorem.^[35]

Once these theories had been developed, it was still several decades until an abstract ring concept emerged. The first axiomatic definition was given byAbraham Fraenkelin 1914.^[35]His definition was mainly the standard axioms: a set with two operations addition, which forms a group (not necessarily commutative), and multiplication, which is associative, distributes over addition, and has an identity element.^[36]In addition, he had two axioms on "regular elements" inspired by work on thep-adic numbers,which excluded now-common rings such as the ring of integers. These allowed Fraenkel to prove that addition was commutative.^[37]Fraenkel's work aimed to transfer Steinitz's 1910 definition of fields over to rings, but it was not connected with the existing work on concrete systems. Masazo Sono's 1917 definition was the first equivalent to the present one.^[38]

In 1920,Emmy Noether,in collaboration with W. Schmeidler, published a paper about thetheory of idealsin which they definedleft and right idealsin aring.The following year she published a landmark paper calledIdealtheorie in Ringbereichen(Ideal theory in rings'), analyzingascending chain conditionswith regard to (mathematical) ideals. The publication gave rise to the term "Noetherian ring",and several other mathematical objects being calledNoetherian.^[39]^[40]Noted algebraistIrving Kaplanskycalled this work "revolutionary";^[39]results which seemed inextricably connected to properties of polynomial rings were shown to follow from a single axiom.^[41]Artin, inspired by Noether's work, came up with thedescending chain condition.These definitions marked the birth of abstract ring theory.^[42]

Early field theory[edit]

In 1801 Gauss introduced theintegers mod p,where p is a prime number. Galois extended this in 1830 tofinite fieldswith $p^{n}$ elements.^[43]In 1871Richard Dedekindintroduced, for a set of real or complex numbers that is closed under the four arithmetic operations,^[44]theGermanwordKörper,which means "body" or "corpus" (to suggest an organically closed entity). The English term "field" was introduced by Moore in 1893.^[45]In 1881Leopold Kroneckerdefined what he called adomain of rationality,which is a field ofrational fractionsin modern terms.^[46]The first clear definition of an abstract field was due toHeinrich Martin Weberin 1893. It was missing the associative law for multiplication, but covered finite fields and the fields of algebraic number theory and algebraic geometry.^[47]In 1910 Steinitz synthesized the knowledge of abstract field theory accumulated so far. He axiomatically defined fields with the modern definition, classified them by theircharacteristic,and proved many theorems commonly seen today.^[48]

Other major areas[edit]

Solving ofsystems of linear equations,which led tolinear algebra^[49]

Modern algebra[edit]

The end of the 19th and the beginning of the 20th century saw a shift in the methodology of mathematics. Abstract algebra emerged around the start of the 20th century, under the namemodern algebra.Its study was part of the drive for moreintellectual rigorin mathematics. Initially, the assumptions in classicalalgebra,on which the whole of mathematics (and major parts of thenatural sciences) depend, took the form ofaxiomatic systems.No longer satisfied with establishing properties of concrete objects, mathematicians started to turn their attention to general theory. Formal definitions of certainalgebraic structuresbegan to emerge in the 19th century. For example, results about various groups of permutations came to be seen as instances of general theorems that concern a general notion of anabstract group.Questions of structure and classification of various mathematical objects came to forefront.^{[citation needed]}

These processes were occurring throughout all of mathematics, but became especially pronounced in algebra. Formal definition through primitive operations and axioms were proposed for many basic algebraic structures, such asgroups,rings,andfields.Hence such things asgroup theoryandring theorytook their places inpure mathematics.The algebraic investigations of general fields byErnst Steinitzand of commutative and then general rings byDavid Hilbert,Emil ArtinandEmmy Noether,building on the work ofErnst Kummer,Leopold KroneckerandRichard Dedekind,who had considered ideals in commutative rings, and ofGeorg FrobeniusandIssai Schur,concerningrepresentation theoryof groups, came to define abstract algebra. These developments of the last quarter of the 19th century and the first quarter of 20th century were systematically exposed inBartel van der Waerden'sModerne Algebra,the two-volumemonographpublished in 1930–1931 that reoriented the idea of algebra fromthe theory of equationstothetheory of algebraic structures.^{[citation needed]}

Basic concepts[edit]

By abstracting away various amounts of detail, mathematicians have defined various algebraic structures that are used in many areas of mathematics. For instance, almost all systems studied aresets,to which the theorems ofset theoryapply. Those sets that have a certain binary operation defined on them formmagmas,to which the concepts concerning magmas, as well those concerning sets, apply. We can add additional constraints on the algebraic structure, such as associativity (to formsemigroups); identity, and inverses (to formgroups); and other more complex structures. With additional structure, more theorems could be proved, but the generality is reduced. The "hierarchy" of algebraic objects (in terms of generality) creates a hierarchy of the corresponding theories: for instance, the theorems ofgroup theorymay be used when studyingrings(algebraic objects that have two binary operations with certain axioms) since a ring is a group over one of its operations. In general there is a balance between the amount of generality and the richness of the theory: more general structures have usually fewernontrivialtheorems and fewer applications.^{[citation needed]}

Examples of algebraic structures with a singlebinary operationare:

Examples involving several operations include:

Branches of abstract algebra[edit]

Group theory[edit]

A group is a set $G$ together with a "group product", a binary operation $\cdot:G\times G\rightarrow G$ .The group satisfies the following defining axioms (c.f.Group (mathematics) § Definition):

Identity:there exists an element $e$ such that, for each element $a$ in $G$ ,it holds that $e\cdot a=a\cdot e=a$ .

Inverse:for each element $a$ of $G$ ,there exists an element $b$ so that $a\cdot b=b\cdot a=e$ .

Associativity:for each triplet of elements $a,b,c$ in $G$ ,it holds that $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ .

Ring theory[edit]

A ring is a set $R$ with twobinary operations,addition: $(x,y)\mapsto x+y,$ and multiplication: $(x,y)\mapsto xy$ satisfying the followingaxioms.

$R$ is acommutative groupunder addition.
$R$ is amonoidunder multiplication.
Multiplication isdistributivewith respect to addition.

Applications[edit]

Because of its generality, abstract algebra is used in many fields of mathematics and science. For instance,algebraic topologyuses algebraic objects to study topologies. ThePoincaré conjecture,proved in 2003, asserts that thefundamental groupof a manifold, which encodes information about connectedness, can be used to determine whether a manifold is a sphere or not.Algebraic number theorystudies various numberringsthat generalize the set of integers. Using tools of algebraic number theory,Andrew WilesprovedFermat's Last Theorem.^{[citation needed]}

In physics, groups are used to represent symmetry operations, and the usage of group theory could simplify differential equations. Ingauge theory,the requirement oflocal symmetrycan be used to deduce the equations describing a system. The groups that describe those symmetries areLie groups,and the study of Lie groups and Lie algebras reveals much about the physical system; for instance, the number offorce carriersin a theory is equal to the dimension of the Lie algebra, and thesebosonsinteract with the force they mediate if the Lie algebra is nonabelian.^[50]

References[edit]

^Finston, David R.; Morandi, Patrick J. (29 August 2014).Abstract Algebra: Structure and Application.Springer. p. 58.ISBN 978-3-319-04498-9.Much of our study of abstract algebra involves an analysis of structures and their operations
^Kleiner 2007,p. 1.
^Kleiner 2007,pp. xi–xii.
^van der Waerden, Bartel Leendert (1949).Modern Algebra. Vol I.Translated by Blum, Fred. New York, N. Y.: Frederick Ungar Publishing Co.MR 0029363.
^Kleiner 2007,p. 41.
^Kleiner 2007,pp. 1–13.
^Euler, Leonard (1748).Introductio in Analysin Infinitorum[Introduction to the Analysis of the Infinite] (in Latin). Vol. 1. Lucerne, Switzerland: Marc Michel Bosquet & Co. p. 104.
^Martinez, Alberto (2014).Negative Math.Princeton University Press. pp. 80–109.
^Kleiner 2007,pp. 13–14.
^Kleiner 2007,pp. 17–22.
^O'Connor, John J.;Robertson, Edmund F.,"The abstract group concept",MacTutor History of Mathematics Archive,University of St Andrews
^Kleiner 2007,p. 23.
^Cayley, A. (1854)."On the theory of groups, as depending on the symbolic equation θⁿ= 1 ".Philosophical Magazine.4th series.7(42): 40–47.doi:10.1080/14786445408647421.
^Kronecker, Leopold (1895). "Auseinandeesetzung einiger eigenschaften der klassenanzahl idealer complexer zahlen" [An exposition of some properties of the class number of ideal complex numbers]. In Hensel, Kurt (ed.).Leopold Kronecker's werke: Herausgegeben auf veranlassung der Königlich preussischen akademie der wissenschaften.Leipzig; Berlin: B.G. Teubner. p. 275.
^Kleiner 2007,p. 27.
^Kleiner 2007,p. 32.
^Kleiner 2007,p. 33.
^^a ^bKleiner 2007,p. 34.
^Frobenius, G. (April 2008) [1887]."Neuer Beweis des Sylowschen Satzes"[New Proof of Sylow's Theorem](PDF).Journal für die reine und angewandte Mathematik(in German).1887(100). Translated by Gutfraind, Sasha: 179–181.doi:10.1515/crll.1887.100.179.S2CID 117970003.
^Kleiner 2007,p. 35.
^Kleiner 2007,pp. 42–43.
^Cockle, James (1848)."On Certain Functions Resembling Quaternions and on a New Imaginary in Algebra".The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science.33.Taylor & Francis: 435–9.doi:10.1080/14786444808646139.
^Cockle, James (1849)."On Systems of Algebra involving more than one Imaginary".The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science.35.Taylor & Francis: 434–7.doi:10.1080/14786444908646384.
^Kleiner 2007,p. 43.
^^a ^bKleiner 2007,pp. 43–47.
^^a ^bKleiner 2007,p. 42.
^Kleiner 2007,p. 48.
^Kleiner 2007,p. 50.
^Kleiner 2007,pp. 51–52.
^Kleiner 2007,p. 54.
^Monna 1975,pp. 55–56, citingHilbert, David (1905), "Über das Dirichletsche Prinzip",Journal für die reine und angewandte Mathematik(in German), vol. 129, pp. 63–67
^Kleiner 2007,pp. 54–57.
^Kleiner 2007,pp. 57–58.
^Gordan, Paul (1868),"Beweis, dass jede Covariante und Invariante einer binären Form eine ganze Funktion mit numerischen Coeffizienten einer endlichen Anzahl solcher Formen ist",Journal für die reine und angewandte Mathematik,vol. 1868, no. 69, pp. 323–354,doi:10.1515/crll.1868.69.323,S2CID 120689164
^^a ^bKleiner 2007,p. 58.
^Frankel, A. (1914) "Über die Teiler der Null und die Zerlegung von Ringen". J. Reine Angew. Math. 145: 139–176
^Corry, Leo (January 2000)."The origins of the definition of abstract rings".Modern Logic.8(1–2): 5–27.ISSN 1047-5982.
^Kleiner 2007,pp. 58–59.
^^a ^bKimberling 1981,p. 18.
^Dick, Auguste(1981),Emmy Noether: 1882–1935,translated by Blocher, H. I.,Birkhäuser,ISBN 3-7643-3019-8,p. 44–45.
^Kleiner 2007,p. 59.
^Kleiner 2007,p. 60.
^Kleiner 2007,p. 70.
^Kleiner 2007,p. 66.
^"Earliest Known Uses of Some of the Words of Mathematics (F)".
^Kleiner 2007,p. 67.
^Kleiner 2007,pp. 72–73.
^Kleiner 2007,pp. 74–76.
^Hart, Roger (2011).The Chinese roots of linear algebra.Baltimore, MD: Johns Hopkins University Press.ISBN 978-0-8018-9958-4.OCLC 794700410.
^Schumm, Bruce (2004),Deep Down Things,Baltimore: Johns Hopkins University Press,ISBN 0-8018-7971-X

Bibliography[edit]

Gray, Jeremy (2018).A history of abstract algebra: from algebraic equations to modern algebra.Springer Undergraduate Mathematics Series. Cham, Switzerland.doi:10.1007/978-3-319-94773-0.ISBN 978-3-319-94773-0.S2CID 125927783.{{cite book}}:CS1 maint: location missing publisher (link)
Kimberling, Clark(1981). "Emmy Noether and Her Influence". In Brewer, James W; Smith, Martha K (eds.).Emmy Noether: A Tribute to Her Life and Work.Marcel Dekker.pp. 3–61.
Kleiner, Israel (2007). Kleiner, Israel (ed.).A history of abstract algebra.Boston, Mass.: Birkhäuser.doi:10.1007/978-0-8176-4685-1.ISBN 978-0-8176-4685-1.
Monna, A. F. (1975),Dirichlet's principle: A mathematical comedy of errors and its influence on the development of analysis,Oosthoek, Scheltema & Holkema,ISBN 978-9031301751

External links[edit]

Charles C. Pinter (1990) [1982]A Book of Abstract Algebra,second edition, fromUniversity of Maryland

[1] Finston, David R.; Morandi, Patrick J. (29 August 2014).Abstract Algebra: Structure and Application.Springer. p. 58.ISBN 978-3-319-04498-9.Much of our study of abstract algebra involves an analysis of structures and their operations

[FOOTNOTEKleiner20071-2] Kleiner 2007,p. 1.

[FOOTNOTEKleiner2007xi–xii-3] Kleiner 2007,pp. xi–xii.

[4] van der Waerden, Bartel Leendert (1949).Modern Algebra. Vol I.Translated by Blum, Fred. New York, N. Y.: Frederick Ungar Publishing Co.MR 0029363.

[FOOTNOTEKleiner200741-5] Kleiner 2007,p. 41.

[FOOTNOTEKleiner20071–13-6] Kleiner 2007,pp. 1–13.

[7] Euler, Leonard (1748).Introductio in Analysin Infinitorum[Introduction to the Analysis of the Infinite] (in Latin). Vol. 1. Lucerne, Switzerland: Marc Michel Bosquet & Co. p. 104.

[8] Martinez, Alberto (2014).Negative Math.Princeton University Press. pp. 80–109.

[FOOTNOTEKleiner200713–14-9] Kleiner 2007,pp. 13–14.

[FOOTNOTEKleiner200717–22-10] Kleiner 2007,pp. 17–22.

[AbsGroup-11] O'Connor, John J.;Robertson, Edmund F.,"The abstract group concept",MacTutor History of Mathematics Archive,University of St Andrews

[FOOTNOTEKleiner200723-12] Kleiner 2007,p. 23.

[13] Cayley, A. (1854)."On the theory of groups, as depending on the symbolic equation θⁿ= 1 ".Philosophical Magazine.4th series.7(42): 40–47.doi:10.1080/14786445408647421.

[14] Kronecker, Leopold (1895). "Auseinandeesetzung einiger eigenschaften der klassenanzahl idealer complexer zahlen" [An exposition of some properties of the class number of ideal complex numbers]. In Hensel, Kurt (ed.).Leopold Kronecker's werke: Herausgegeben auf veranlassung der Königlich preussischen akademie der wissenschaften.Leipzig; Berlin: B.G. Teubner. p. 275.

[FOOTNOTEKleiner200727-15] Kleiner 2007,p. 27.

[FOOTNOTEKleiner200732-16] Kleiner 2007,p. 32.

[FOOTNOTEKleiner200733-17] Kleiner 2007,p. 33.

[FOOTNOTEKleiner200734-18] Kleiner 2007,p. 34.

[19] Frobenius, G. (April 2008) [1887]."Neuer Beweis des Sylowschen Satzes"[New Proof of Sylow's Theorem](PDF).Journal für die reine und angewandte Mathematik(in German).1887(100). Translated by Gutfraind, Sasha: 179–181.doi:10.1515/crll.1887.100.179.S2CID 117970003.

[FOOTNOTEKleiner200735-20] Kleiner 2007,p. 35.

[FOOTNOTEKleiner200742–43-21] Kleiner 2007,pp. 42–43.

[22] Cockle, James (1848)."On Certain Functions Resembling Quaternions and on a New Imaginary in Algebra".The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science.33.Taylor & Francis: 435–9.doi:10.1080/14786444808646139.

[23] Cockle, James (1849)."On Systems of Algebra involving more than one Imaginary".The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science.35.Taylor & Francis: 434–7.doi:10.1080/14786444908646384.

[FOOTNOTEKleiner200743-24] Kleiner 2007,p. 43.

[FOOTNOTEKleiner200743–47-25] Kleiner 2007,pp. 43–47.

[FOOTNOTEKleiner200742-26] Kleiner 2007,p. 42.

[FOOTNOTEKleiner200748-27] Kleiner 2007,p. 48.

[FOOTNOTEKleiner200750-28] Kleiner 2007,p. 50.

[FOOTNOTEKleiner200751–52-29] Kleiner 2007,pp. 51–52.

[FOOTNOTEKleiner200754-30] Kleiner 2007,p. 54.

[31] Monna 1975,pp. 55–56, citingHilbert, David (1905), "Über das Dirichletsche Prinzip",Journal für die reine und angewandte Mathematik(in German), vol. 129, pp. 63–67

[FOOTNOTEKleiner200754–57-32] Kleiner 2007,pp. 54–57.

[FOOTNOTEKleiner200757–58-33] Kleiner 2007,pp. 57–58.

[34] Gordan, Paul (1868),"Beweis, dass jede Covariante und Invariante einer binären Form eine ganze Funktion mit numerischen Coeffizienten einer endlichen Anzahl solcher Formen ist",Journal für die reine und angewandte Mathematik,vol. 1868, no. 69, pp. 323–354,doi:10.1515/crll.1868.69.323,S2CID 120689164

[FOOTNOTEKleiner200758-35] Kleiner 2007,p. 58.

[36] Frankel, A. (1914) "Über die Teiler der Null und die Zerlegung von Ringen". J. Reine Angew. Math. 145: 139–176

[37] Corry, Leo (January 2000)."The origins of the definition of abstract rings".Modern Logic.8(1–2): 5–27.ISSN 1047-5982.

[FOOTNOTEKleiner200758–59-38] Kleiner 2007,pp. 58–59.

[FOOTNOTEKimberling198118-39] Kimberling 1981,p. 18.

[40] Dick, Auguste(1981),Emmy Noether: 1882–1935,translated by Blocher, H. I.,Birkhäuser,ISBN 3-7643-3019-8,p. 44–45.

[FOOTNOTEKleiner200759-41] Kleiner 2007,p. 59.

[FOOTNOTEKleiner200760-42] Kleiner 2007,p. 60.

[FOOTNOTEKleiner200770-43] Kleiner 2007,p. 70.

[FOOTNOTEKleiner200766-44] Kleiner 2007,p. 66.

[45] "Earliest Known Uses of Some of the Words of Mathematics (F)".

[FOOTNOTEKleiner200767-46] Kleiner 2007,p. 67.

[FOOTNOTEKleiner200772–73-47] Kleiner 2007,pp. 72–73.

[FOOTNOTEKleiner200774–76-48] Kleiner 2007,pp. 74–76.

[49] Hart, Roger (2011).The Chinese roots of linear algebra.Baltimore, MD: Johns Hopkins University Press.ISBN 978-0-8018-9958-4.OCLC 794700410.

[50] Schumm, Bruce (2004),Deep Down Things,Baltimore: Johns Hopkins University Press,ISBN 0-8018-7971-X

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v t e Algebra
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