In that setting one can use birational geometry, techniques from number theory, Galois theory and commutative algebra, and close analogues of the methods of algebraic topology, all in an integrated way.

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

In part, the growth was spurred by new connections and applications to other fields, ranging from algebra to probability, from functional analysis to number theory, etc.

*: This number arises so often in numerical linear algebra that it is given a name, the condition number of a matrix.

*: The condition number computed with this norm is generally larger than the condition number computed with square-summable sequences, but it can be evaluated more easily ( and this is often the only measurable condition number, when the problem to solve involves a non-linear algebra, for example when approximating irrational and transcendental functions or numbers with numerical methods.

The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra.

Diophantus is often called “ the father of algebra " because he contributed greatly to number theory, mathematical notation, and because Arithmetica contains the earliest known use of syncopated notation.

Much of the Elements states results of what are now called algebra and number theory, couched in geometrical language.

In abstract algebra, a finite field or Galois field ( so named in honor of Évariste Galois ) is a field that contains a finite number of elements.

A GUT model basically consists of a gauge group which is a compact Lie group, a connection form for that Lie group, a Yang-Mills action for that connection given by an invariant symmetric bilinear form over its Lie algebra ( which is specified by a coupling constant for each factor ), a Higgs sector consisting of a number of scalar fields taking on values within real / complex representations of the Lie group and chiral Weyl fermions taking on values within a complex rep of the Lie group.

He worked on a great variety of mathematical topics, including series, number theory, mathematical analysis, geometry, algebra, combinatorics, and probability.

The concept of idempotence arises in a number of places in abstract algebra ( in particular, in the theory of projectors and closure operators ) and functional programming ( in which it is connected to the property of referential transparency ).

Euler worked in almost all areas of mathematics: geometry, infinitesimal calculus, trigonometry, algebra, and number theory, as well as continuum physics, lunar theory and other areas of physics.

The Lie algebra of any compact Lie group ( very roughly: one for which the symmetries form a bounded set ) can be decomposed as a direct sum of an abelian Lie algebra and some number of simple ones.

For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations.

Mathematicians do research in fields such as logic, set theory, category theory, abstract algebra, number theory, analysis, geometry, topology, dynamical systems, combinatorics, game theory, information theory, numerical analysis, optimization, computation, probability and statistics.

While Babylonian number theory — or what survives of Babylonian mathematics that can be called thus — consists of this single, striking fragment, Babylonian algebra ( in the secondary-school sense of " algebra ") was exceptionally well developed.

Both these approaches apply to other areas of mathematics beyond analysis, including number theory, algebra and topology.

Prime ideals, which generalize prime elements in the sense that the principal ideal generated by a prime element is a prime ideal, are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry.

We have chosen to give it at the end of the section since it deals with differential equations and thus is not purely linear algebra.

has no zero in F. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed.

In the context of abstract algebra, for example, a mathematical object is an algebraic structure such as a group, ring, or vector space.

* In linear algebra, an endomorphism of a vector space V is a linear operator V → V. An automorphism is an invertible linear operator on V. When the vector space is finite-dimensional, the automorphism group of V is the same as the general linear group, GL ( V ).

The same definition holds in any unital ring or algebra where a is any invertible element.

In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry.

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry.

His notion of abelian category is now the basic object of study in homological algebra.

With the existence of an alpha channel, it is possible to express compositing image operations, using a compositing algebra.

Explicitly, is an associative algebra homomorphism if

* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.

* An example of a non-unitary associative algebra is given by the set of all functions f: R → R whose limit as x nears infinity is zero.

* Any ring of characteristic n is a ( Z / nZ )- algebra in the same way.

* Any ring A is an algebra over its center Z ( A ), or over any subring of its center.

* Any commutative ring R is an algebra over itself, or any subring of R.

Other important Arabic astrologers include Albumasur and Al Khwarizmi, the Persian mathematician, astronomer and astrologer, who is considered the father of algebra and the algorithm.

Notice that C < sub > b </ sub >( X ) is the multiplier algebra of C < sub > 0 </ sub >( X ).

His view of arithmetical algebra is as follows: " In arithmetical algebra we consider symbols as representing numbers, and the operations to which they are submitted as included in the same definitions as in common arithmetic ; the signs and denote the operations of addition and subtraction in their ordinary meaning only, and those operations are considered as impossible in all cases where the symbols subjected to them possess values which would render them so in case they were replaced by digital numbers ; thus in expressions such as we must suppose and to be quantities of the same kind ; in others, like, we must suppose greater than and therefore homogeneous with it ; in products and quotients, like and we must suppose the multiplier and divisor to be abstract numbers ; all results whatsoever, including negative quantities, which are not strictly deducible as legitimate conclusions from the definitions of the several operations must be rejected as impossible, or as foreign to the science.

This extends uniquely to a representation of its multiplier algebra and therefore a strongly continuous unitary representation U < sub > g </ sub >.

In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F, the ring of polynomials in the variable x with coefficients in F.

In his 1799 doctorate in absentia, A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

* 1799: Doctoral dissertation on the Fundamental theorem of algebra, with the title: Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse (" New proof of the theorem that every integral algebraic function of one variable can be resolved into real factors ( i. e., polynomials ) of the first or second degree ")

It is often convenient to express the theory using the algebra of random variables: thus if X is used to denote a random variable corresponding to the observed data, the estimator ( itself treated as a random variable ) is symbolised as a function of that random variable,.

Especially, the fact that the integers and any polynomial ring in one variable over a field are Euclidean domains such that the Euclidean division is easily computable is of basic importance in computer algebra.

By mathematical analogy: A metasyntactic variable is a word that is a variable for other words, just as in algebra letters are used as variables for numbers.

In probability theory, the sigma algebra often represents the set of available information, and a function ( in this context a random variable ) is measurable if and only if it represents an outcome that is knowable based on the available information.

Sometimes, more general variants of the notion of fuzzy set are used, with membership functions taking values in a ( fixed or variable ) algebra or structure of a given kind ; usually it is required that be at least a poset or lattice.

In the paper of Roman and Rota cited below, the umbral calculus is characterized as the study of the umbral algebra, defined as the algebra of linear functionals on the vector space of polynomials in a variable x, with a product L < sub > 1 </ sub > L < sub > 2 </ sub > of linear functionals defined by

In linear algebra, the characteristic equation ( or secular equation ) of a square matrix A is the equation in one variable λ

In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable.

The infinitesimal differentials of single variable calculus become differential forms in multivariate calculus, and their manipulation is done with exterior algebra.

If L is abelian ( that is, the bracket is always 0 ), then U ( L ) is commutative ; if a basis of the vector space L has been chosen, then U ( L ) can be identified with the polynomial algebra over K, with one variable per basis element.

* Coß ( manuscript 1524, printed in 1992 ): The algebra textbook is named after the common name for the unknown variable in the German Middle Ages, and it establishes the connection between medieval and modern algebra.

In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients ( in one variable ),

At that time Klein was dealing with deep problems of algebra and theory of functions of a complex variable.

Even the case of representations of the polynomial algebra in a single variable are of interest – this is denoted by and is used in understanding the structure of a single linear operator on a finite-dimensional vector space.

The partial derivative with respect to a variable is an R-derivation on the algebra of real-valued differentiable functions on R < sup > n </ sup >.