Abstract algebra

In

Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory gives a unified framework to study properties and constructions that are similar for various structures.

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.

History

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]

Elementary algebra

The study of polynomial equations or

arithmetical algebra

. Whereas in arithmetical algebra ${\displaystyle a-b}$ is restricted to ${\displaystyle a\geq b}$, in symbolical algebra all rules of operations hold with no restrictions. Using this Peacock could show laws such as ${\displaystyle (-a)(-b)=ab}$, by letting ${\displaystyle a=0,c=0}$ in ${\displaystyle (a-b)(c-d)=ac+bd-ad-bc}$. Peacock used what he termed the

For example, ${\displaystyle {\sqrt {a}}{\sqrt {b}}={\sqrt {ab}}}$ holds for the nonnegative

Early group theory

Several areas of mathematics led to the study of groups. Lagrange's 1770 study of the solutions of the quintic equation led to the

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 paper On the theory of groups defined a group as a set with an associative composition operation and the identity 1, today called a monoid.^[13] In 1870 Kronecker defined an abstract binary operation that was closed, commutative, associative, and had the left cancellation property ${\displaystyle b\neq c\to a\cdot b\neq a\cdot c}$,^[14] similar to the modern laws for a finite abelian 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 Dyck in 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,

Noncommutative ring theory began with extensions of the complex numbers to

Once there were sufficient examples, it remained to classify them. 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. He defined nilpotent and idempotent elements and proved that any algebra contains one or the other. He also defined the Peirce decomposition. Frobenius in 1878 and Charles Sanders Peirce in 1881 independently proved that the only finite-dimensional division algebras over ${\displaystyle \mathbb {R} }$ were the real numbers, the complex numbers, and the quaternions. In the 1880s Killing and Cartan showed that semisimple Lie algebras could 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 ${\displaystyle \mathbb {R} }$ or ${\displaystyle \mathbb {C} }$ uniquely decomposes into the

were UFDs, yet as Kummer pointed out, ${\displaystyle \mathbb {Q} (\zeta _{23}))}$ was not a UFD.

In the 1850s, Riemann introduced the fundamental concept of a Riemann surface. Riemann's methods relied on an assumption he called Dirichlet's principle,^[30] which in 1870 was questioned by Weierstrass. Much later, in 1900, Hilbert justified Riemann's approach by developing the direct method in the calculus of variations.^[31] In the 1860s and 1870s, Clebsch, Gordan, Brill, and especially M. Noether studied algebraic functions and 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 ${\displaystyle \mathbb {R} [x,y]}$, 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 of

Once these theories had been developed, it was still several decades until an abstract ring concept emerged. The first axiomatic definition was given by Abraham Fraenkel in 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 the p-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,

In 1801 Gauss introduced the

with ${\displaystyle p^{n}}$ elements.

• Solving of
  systems of linear equations, which led to linear algebra^[49]

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

Examples of algebraic structures with a single binary operation are:

• Magma
• Quasigroup
• Monoid
• Semigroup
• Group

Examples involving several operations include:

Branches of abstract algebra

Group theory

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

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

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

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

Ring theory

A ring is a set ${\displaystyle R}$ with two binary operations, addition: ${\displaystyle (x,y)\mapsto x+y,}$ and multiplication: ${\displaystyle (x,y)\mapsto xy}$ satisfying the following axioms.

• ${\displaystyle R}$ is a
  commutative group
  under addition.
• ${\displaystyle R}$ is a monoid under multiplication.
• Multiplication is
  distributive
  with respect to addition.

Applications

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

• Coding theory
• Group theory
• List of publications in abstract algebra

References

Bibliography

• Gray, Jeremy (2018). A history of abstract algebra: from algebraic equations to modern algebra. Springer Undergraduate Mathematics Series. Cham, Switzerland.
  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.
  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

Further reading

• Allenby, R. B. J. T. (1991), Rings, Fields and Groups, Butterworth-Heinemann,
  ISBN 978-0-340-54440-2
• ISBN 978-0-89871-510-1
• Burris, Stanley N.; Sankappanavar, H. P. (1999) [1981], A Course in Universal Algebra
• Gilbert, Jimmie; Gilbert, Linda (2005), Elements of Modern Algebra, Thomson Brooks/Cole,
  ISBN 978-0-534-40264-8
• MR 1878556
• Sethuraman, B. A. (1996), Rings, Fields, Vector Spaces, and Group Theory: An Introduction to Abstract Algebra via Geometric Constructibility, Berlin, New York:
  ISBN 978-0-387-94848-5
• Whitehead, C. (2002), Guide to Abstract Algebra (2nd ed.), Houndmills: Palgrave,
  ISBN 978-0-333-79447-0
• W. Keith Nicholson (2012) Introduction to Abstract Algebra, 4th edition,
  ISBN 978-1-118-13535-8
  .
• John R. Durbin (1992) Modern Algebra : an introduction, John Wiley & Sons

External links

Wikibooks has more on the topic of: Abstract algebra

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