Abstract algebra
[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.
This word problem stage is classified as rhetorical algebra and was the dominant approach up to the 16th century.
Peacock used what he termed the principle of the permanence of equivalent forms to justify his argument, but his reasoning suffered from the problem of induction.
[13] In 1870 Kronecker defined an abstract binary operation that was closed, commutative, associative, and had the left cancellation property
Dedekind and Miller independently characterized Hamiltonian groups and introduced the notion of the commutator of two elements.
Burnside, Frobenius, and Molien created the representation theory of finite groups at the end of the nineteenth century.
In addition Cayley introduced group algebras over the real and complex numbers in 1854 and square matrices in two papers of 1855 and 1858.
In an 1870 monograph, Benjamin Peirce classified the more than 150 hypercomplex number systems of dimension below 6, and gave an explicit definition of an associative algebra.
Frobenius in 1878 and Charles Sanders Peirce in 1881 independently proved that the only finite-dimensional division algebras over
Inspired by this, in the 1890s Cartan, Frobenius, and Molien proved (independently) that a finite-dimensional associative algebra over
[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.
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,
Riemann's methods relied on an assumption he called Dirichlet's principle,[30] which in 1870 was questioned by Weierstrass.
In 1882 Dedekind and Weber, in analogy with Dedekind's earlier work on algebraic number theory, created a theory of algebraic function fields which allowed the first rigorous definition of a Riemann surface and a rigorous proof of the Riemann–Roch theorem.
Kronecker in the 1880s, Hilbert in 1890, Lasker in 1905, and Macauley in 1913 further investigated the ideals of polynomial rings implicit in E. Noether's work.
[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.
[39][40] Noted algebraist Irving Kaplansky called this work "revolutionary";[39] results which seemed inextricably connected to properties of polynomial rings were shown to follow from a single axiom.
He axiomatically defined fields with the modern definition, classified them by their characteristic, and proved many theorems commonly seen today.
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 through primitive operations and axioms were proposed for many basic algebraic structures, such as groups, rings, and fields.
The algebraic investigations of general fields by Ernst Steinitz and of commutative and then general rings by David Hilbert, Emil Artin and Emmy Noether, building on the work of Ernst Kummer, Leopold Kronecker and Richard Dedekind, who had considered ideals in commutative rings, and of Georg Frobenius and Issai Schur, concerning representation theory of groups, came to define abstract algebra.
[51] By abstracting away various amounts of detail, mathematicians have defined various algebraic structures that are used in many areas of mathematics.
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.
Using tools of algebraic number theory, Andrew Wiles proved Fermat's Last Theorem.
