Early field theory

 

Early field theory[edit]

Main article: Field (mathematics) § History

In 1801 Gauss introduced the integers mod p, where p is a prime number. Galois extended this in 1830 to finite fields with ${\displaystyle {�}^{�}}$ 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]

Other major areas[edit]

• Solving of systems of linear equations, which led to linear 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 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.

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 as groups, rings, and fields. Hence such things as group theory and ring theory took their places in pure mathematics. 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 up 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. These developments of the last quarter of the 19th century and the first quarter of 20th century were systematically exposed in Bartel van der Waerden's Moderne Algebra, the two-volume monograph published in 1930–1931 that forever changed for the mathematical world the meaning of the word algebra from the theory of equations to the theory of algebraic structures.

Basic concepts[edit]

Main article: Algebraic structure

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.

Algebraic structures between magmas and groups. For example, monoids are semigroups with identity.

Examples of algebraic structures with a single binary operation are:

• Magma
• Quasigroup
• Monoid
• Semigroup
• Group

Examples involving several operations include:

• Ring
• Field
• Module
• Vector space
• Algebra over a field
• Associative algebra
• Lie algebra
• Lattice
• Boolean algebra

Branches of abstract algebra[edit]

Group theory[edit]

Main article: Group theory

A group is a set ${\displaystyle �}$ together with a "group product", a binary operation ${\displaystyle \cdot :�\times �\to �}$. The group satisfies the following defining axioms:

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

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

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