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Post-Modern Algebra

In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures.[1] Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Presently, the term "abstract algebra" is typically used for naming courses in mathematical education, and is rarely used in advanced mathematics.

Post-Modern Algebra

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Universal algebra is 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 the variety of groups.

Before the nineteenth century, algebra was defined as the study of polynomials.[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 of algebraic 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 formal axiomatic definitions of various algebraic structures such 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's Moderne Algebra,[4] which start each chapter with a formal definition of a structure and then follow it with concrete examples.[5]

The study of polynomial equations or algebraic equations has a long history. Circa 1700 BC, the Babylonians were able to solve quadratic equations specified as word problems. This word problem stage is classified as rhetorical algebra and was the dominant approach up to the 16th century. Muhammad ibn Mūsā al-Khwārizmī originated the word "algebra" in 830 AD, but his work was entirely rhetorical algebra. Fully symbolic algebra did not appear until François Viète's 1591 New Algebra, and even this had some spelled out words that were given symbols in Descartes's 1637 La Géométrie.[6] The formal study of solving symbolic equations led Leonhard Euler to accept what were then considered "nonsense" roots such as negative numbers and imaginary 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]

Noncommutative ring theory began with extensions of the complex numbers to hypercomplex numbers, specifically William Rowan Hamilton's quaternions in 1843. Many other number systems followed shortly. In 1844, Hamilton presented biquaternions, Cayley introduced octonions, and Grassman introduced exterior algebras.[21] James Cockle presented tessarines in 1848[22] and coquaternions in 1849.[23] William Kingdon Clifford introduced split-biquaternions in 1873. In addition Cayley introduced group algebras over the real and complex numbers in 1854 and square matrices in two papers of 1855 and 1858.[24]

For commutative rings, several areas together led to commutative ring theory.[26] In two papers in 1828 and 1832, Gauss formulated the Gaussian integers and showed that they form a unique factorization domain (UFD) and proved the biquadratic reciprocity law. Jacobi and Eisenstein at around the same time proved a cubic reciprocity law for the Eisenstein integers.[25] The study of Fermat's last theorem led to the algebraic integers. In 1847, Gabriel Lamé thought he had proven FLT, but his proof was faulty as he assumed all the cyclotomic fields were UFDs, yet as Kummer pointed out, Q ( ζ 23 ) ) \displaystyle \mathbb Q (\zeta _23)) was not a UFD.[27] In 1846 and 1847 Kummer introduced ideal numbers and 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 of prime ideals, a precursor of the theory of Dedekind domains. Overall, Dedekind's work created the subject of algebraic number theory.[29]

In 1801 Gauss introduced binary quadratic forms over the integers and defined their equivalence. He further defined the discriminant of these forms, which is an invariant of a binary form. Between the 1860s and 1890s invariant theory developed and became a major field of algebra. Cayley, Sylvester, Gordan and others found the Jacobian and the Hessian for binary quartic forms and cubic forms.[33] In 1868 Gordan proved that the graded algebra of 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 to Hilbert's basis theorem.[35]

In 1801 Gauss introduced the integers mod p, where p is a prime number. Galois extended this in 1830 to finite fields with p n \displaystyle p^n elements.[43] In 1871 Richard Dedekind introduced, for a set of real or complex numbers that is closed under the four arithmetic operations,[44] the German word Körper, which means "body" or "corpus" (to suggest an organically closed entity). The English term "field" was introduced by Moore in 1893.[45] In 1881 Leopold Kronecker defined what he called a domain of rationality, which is a field of rational fractions in modern terms. [46] The first clear definition of an abstract field was due to Heinrich Martin Weber in 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 their characteristic, and proved many theorems commonly seen today.[48]

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 name modern algebra. Its study was part of the drive for more intellectual rigor in mathematics. Initially, the assumptions in classical algebra, on which the whole of mathematics (and major parts of the natural sciences) depend, took the form of axiomatic systems. No longer satisfied with establishing properties of concrete objects, mathematicians started to turn their attention to general theory. Formal definitions of certain algebraic structures began 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 an abstract group. Questions of structure and classification of various mathematical objects came to forefront.[citation needed]

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 are sets, to which the theorems of set theory apply. Those sets that have a certain binary operation defined on them form magmas, 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 form semigroups); identity, and inverses (to form groups); 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 of group theory may be used when studying rings (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 fewer nontrivial theorems and fewer applications.[citation needed]

Because of its generality, abstract algebra is used in many fields of mathematics and science. For instance, algebraic topology uses algebraic objects to study topologies. The Poincaré conjecture, proved in 2003, asserts that the fundamental group of a manifold, which encodes information about connectedness, can be used to determine whether a manifold is a sphere or not. Algebraic number theory studies various number rings that generalize the set of integers. Using tools of algebraic number theory, Andrew Wiles proved Fermat's Last Theorem.[citation needed]

In physics, groups are used to represent symmetry operations, and the usage of group theory could simplify differential equations. In gauge theory, the requirement of local symmetry can be used to deduce the equations describing a system. The groups that describe those symmetries are Lie groups, and the study of Lie groups and Lie algebras reveals much about the physical system; for instance, the number of force carriers in a theory is equal to the dimension of the Lie algebra, and these bosons interact with the force they mediate if the Lie algebra is nonabelian.[50]

(1) Mathematics education, through both the NCTM Standards and calculus reform, has the goal of making mathematics accessible to all students. But I wonder, after we pare away whatever of mathematics is not accessible to everyone, whether what remains will still be mathematics. If, in fact, we hope in vain that the masses can master mathematics, then perhaps calculus (or algebra) does have a legitimate role as a filter for some students, as well as being a pump for some others. If we make calculus (or algebra) easy enough for everybody, we might well find that we have so dumbed it down that we have dumbed it away. 041b061a72

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