commutative algebra

All polynomials have an expanded form, in which the distributive law has been used to remove all parentheses. (Some polynomials also have a factored form, in which parentheses appear.) In expanded form, a term of a polynomial is a part of the polynomial that includes only the operation of multiplication. Every polynomial in expanded form is a sum of terms (where subtraction is carried out by addition of negative numbers). Polynomials are classified by their degree and number of variables. The degree of a term in a polynomial is the su ...

See also:

Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables

Read more here: » Polynomial: Encyclopedia II - Polynomial - Elementary properties of polynomials

In a vector space, the set of scalars forms a field and acts on the vectors by scalar multiplication, subject to certain formal laws such as the distributive law. In a module, the scalars need only be a ring, so the module concept represents a significant generalization. Much of the theory of modules consists of extending as many as possible of the desirable properties of vector spaces to the realm of modules over a "well-behaved" ring, such as a principal ideal domain. However, modules can be quite a bit more complicate ...

See also:

Module mathematics, Module mathematics - Motivation, Module mathematics - Definition, Module mathematics - Examples, Module mathematics - Submodules and homomorphisms, Module mathematics - Types of modules, Module mathematics - Relation to representation theory, Module mathematics - Generalizations

Read more here: » Module mathematics: Encyclopedia II - Module mathematics - Motivation

The algebraic geometers of the Italian school had often used the somewhat foggy concept of "generic point" when proving statements about algebraic varieties. What is true for the generic point is true for all points of the variety except a small number of special points. In the 1920s, Emmy Noether had first suggested a way to clarify the concept: start with the coordinate ring of the variety (the ring of all polynomial functions defined on the variety); the maximal ideals of this ring will correspond to ordinary points of the variety (under ...

See also:

Scheme mathematics, Scheme mathematics - History and motivation, Scheme mathematics - Definitions, Scheme mathematics - The category of schemes, Scheme mathematics - Types of schemes, Scheme mathematics - O_X modules

Read more here: » Scheme mathematics: Encyclopedia II - Scheme mathematics - History and motivation

Gröbner basis - Deciding equality of ideals. Reduced Gröbner bases can be shown to be unique for any given ideal and monomial ordering, and are also often computable in practice. Thus one can determine if two ideals are equal by looking at their reduced Gröbner bases. Gröbner basis - Deciding membership of ideals. The reduction of a polynomial f by the multivariate division algorithm for an ideal using a Gröbner basis will yield 0 if and only if fSee also:

Gröbner basis, Gröbner basis - Formal definition, Gröbner basis - Properties and applications of Gröbner bases, Gröbner basis - Deciding equality of ideals, Gröbner basis - Deciding membership of ideals, Gröbner basis - Elimination property, Gröbner basis - Solving equations, Gröbner basis - Conversion of parametric equations, Gröbner basis - Intersecting ideals

Read more here: » Gröbner basis: Encyclopedia II - Gröbner basis - Properties and applications of Gröbner bases

In complex analysis, there are four kinds of singularity, to be described below. Suppose U is an open subset of the complex numbers C, a is an element of U, and f is a holomorphic function defined on U \ {a}. The point a is a removable singularity of f if there exists a holomorphic function g defined on all of U such that f(z) = g(z) for all z in U − {a}. The point a is a pole ...

See also:

Mathematical singularity, Mathematical singularity - Complex analysis, Mathematical singularity - From the point of view of dynamics, Mathematical singularity - Algebraic geometry and commutative algebra

Read more here: » Mathematical singularity: Encyclopedia II - Mathematical singularity - Complex analysis

The tangent bundle comes equipped with a natural topology (not the disjoint union topology) and smooth structure so as to make it into a manifold it its own right. The dimension of T(M) is twice the dimension of M. Each tangent space of an n-dimensional vector space is an n-dimensional vector space. As a set then, T(M) is isomorphic to M × Rn. As a manifold, however, T(M) is not always diffeomorphic to the product manifold M à ...

See also:

Tangent bundle, Tangent bundle - Topology and smooth structure, Tangent bundle - Examples, Tangent bundle - Vector fields

Read more here: » Tangent bundle: Encyclopedia II - Tangent bundle - Topology and smooth structure

If M and N are two smooth manifolds, how do we define the jet of a function ? We could perhaps attempt to define such a jet by using local coordinates on M and N. The disadvantage of this is that jets cannot thus be defined in an equivariant fashion. Jets do not transform as tensors. In fact, jets of functions between two manifolds belong to a Jet bundle. This section begins by introducing the notion of jets of functions from the real line to a manifold. It proves that such jets form a vector bundle, analog ...

See also:

Jet mathematics, Jet mathematics - Jets of functions between Euclidean spaces, Jet mathematics - Example: One-dimensional case, Jet mathematics - Example: Mappings from one Euclidean space to another, Jet mathematics - Example: Algebraic properties of jets, Jet mathematics - Jets at a point in Euclidean space: Rigorous definitions, Jet mathematics - An analytic definition, Jet mathematics - An algebro-geometric definition, Jet mathematics - Taylor's theorem, Jet mathematics - Jet spaces from a point to a point, Jet mathematics - Jets of functions between two manifolds, Jet mathematics - Jets of functions from the real line to a manifold, Jet mathematics - Jets of functions from a manifold to a manifold, Jet mathematics - Jets of sections, Jet mathematics - Differential operators between vector bundles

Read more here: » Jet mathematics: Encyclopedia II - Jet mathematics - Jets of functions between two manifolds

First we start with a field k. In classical algebraic geometry, this field was always C, the complex numbers, but many of the same results are true if we assume only that k is algebraically closed. We define , called the affine n-space over k, to be kn. The purpose of this apparently superfluous notation is to emphasize that one `forgets' the vector space structure that kn carries. Abstractly speaking, is, for the moment, just a collection of points. Henceforth we will drop the k in and instea ...

See also:

Algebraic geometry, Algebraic geometry - Zeroes of simultaneous polynomials, Algebraic geometry - Affine varieties, Algebraic geometry - Regular functions, Algebraic geometry - The category of affine varieties, Algebraic geometry - Projective space, Algebraic geometry - The modern viewpoint, Algebraic geometry - Notes and history

Read more here: » Algebraic geometry: Encyclopedia II - Algebraic geometry - Affine varieties

The fundamental theorem of arithmetic states that every positive integer larger than 1 can be written as a product of primes in a unique way, i.e. unique except for the order. Primes are thus the "basic building blocks" of the natural numbers (The proof of this is below). For example, we can write and any other such factorization of 23244 will be identical except for the order of the factors. See prime fac ...

See also:

Prime number, Prime number - Representing natural numbers as products of primes, Prime number - How many prime numbers are there?, Prime number - Finding prime numbers, Prime number - Some properties of primes, Prime number - Open questions, Prime number - The largest known prime, Prime number - Applications, Prime number - Primality tests, Prime number - Some special types of primes, Prime number - Prime gaps, Prime number - Formulae yielding prime numbers, Prime number - Generalizations, Prime number - Prime elements in rings, Prime number - Prime ideals, Prime number - Primes in valuation theory, Prime number - Quotes, Prime number - Primes in pop culture

Read more here: » Prime number: Encyclopedia II - Prime number - Representing natural numbers as products of primes

From a very young age he was an outstanding figure in the school of Henri Cartan, working on algebraic topology, several complex variables and then commutative algebra and algebraic geometry, in the context of sheaf theory and homological algebra techniques. Serre's thesis refers to his dissertation on the Leray-Serre spectral sequence associated to a fibration. In his speech at the Fields Medal award ceremony in 1954, Hermann Weyl praised Serre in apparently extravagant terms, and also made the point that the award ...

See also:

Jean-Pierre Serre, Jean-Pierre Serre - Life and career, Jean-Pierre Serre - Early work, Jean-Pierre Serre - Foundational work in algebraic geometry and the Weil conjectures, Jean-Pierre Serre - Other work, Jean-Pierre Serre - Awards, Jean-Pierre Serre - Works, Jean-Pierre Serre - External link

Read more here: » Jean-Pierre Serre: Encyclopedia II - Jean-Pierre Serre - Early work

Let a and b be in the radical of an ideal I. Then, for some positive integers m and n, an and bm are in I. We will show that a + b is in I. Use the binomial theorem to expand (a+b)n+m−1: For each i, exactly one of the following conditions will hold: iSee also:

Radical of an ideal, Radical of an ideal - Definition, Radical of an ideal - Examples, Radical of an ideal - Proof that the radical is an ideal, Radical of an ideal - The nilradical of a ring, Radical of an ideal - Jacobson radicals, Radical of an ideal - Properties, Radical of an ideal - Applications

Read more here: » Radical of an ideal: Encyclopedia II - Radical of an ideal - Proof that the radical is an ideal

Aiming at a completely self-contained treatment of most of modern mathematics based on set theory, the group produced the following volumes (with the original French titles in brackets): I Set theory (Théorie des ensembles) II Algebra (Algèbre) III Topology (Topologie générale) IV Functions of one real variable (Fonctions d'une variable réelle) V Topological vector spaces (Espaces vectoriels topologiques ...

See also:

Nicolas Bourbaki, Nicolas Bourbaki - Books by Bourbaki, Nicolas Bourbaki - Influence on mathematics in general, Nicolas Bourbaki - The group, Nicolas Bourbaki - The Bourbaki perspective and its limitations, Nicolas Bourbaki - Dieudonné as speaker for Bourbaki, Nicolas Bourbaki - The Bourbachique influence: education institutions trends

Read more here: » Nicolas Bourbaki: Encyclopedia II - Nicolas Bourbaki - Books by Bourbaki

The most commonly used differential operator is the action of taking the derivative itself. Common notations for this operator include: where the variable one is differentiating to is clear, and where the variable is declared explicitly. First derivatives are signified as above, but when taking higher, n-th derivatives, the following alterations are useful: The D notation's use and creation is credited to Oliver Heaviside, who considered differential operators of the form See also:

Differential operator, Differential operator - Notations, Differential operator - Adjoint of an operator, Differential operator - Properties of differential operators, Differential operator - Several variables, Differential operator - Coordinate-independent description, Differential operator - Examples

Read more here: » Differential operator: Encyclopedia II - Differential operator - Notations

All polynomials have an expanded form, in which the distributive law has been used to remove all parentheses. (Some polynomials also have a factored form, in which parentheses appear.) In expanded form, a term of a polynomial is a part of the polynomial that includes only the operation of multiplication (where whole number powers are viewed as repeated multiplication). Every polynomial in expanded form is a sum of terms ...

See also:

Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables

Read more here: » Polynomial: Encyclopedia II - Polynomial - Elementary properties of polynomials

The underlying drive, in Weil and Chevalley at least, was the perceived need for French mathematics to absorb the best ideas of the Göttingen school and the German algebraists. It is fairly clear that the Bourbaki point of view, while encyclopedic, was never intended as neutral. Quite the opposite, really: more a question of trying to make a consistent whole out of some enthusiasms, for example for Hilbert's legacy, with emphasis on formalism and axiomatics. But always through a transforming proces ...

See also:

Nicolas Bourbaki, Nicolas Bourbaki - Books by Bourbaki, Nicolas Bourbaki - Influence on mathematics in general, Nicolas Bourbaki - The group, Nicolas Bourbaki - The Bourbaki perspective, and its limitations, Nicolas Bourbaki - Dieudonné as speaker for Bourbaki, Nicolas Bourbaki - The Bourbachique influence: education, institutions, trends

Read more here: » Nicolas Bourbaki: Encyclopedia II - Nicolas Bourbaki - The Bourbaki perspective, and its limitations

The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. Prime number - Prime elements in rings. One can define prime elements and irreducible elements in any integral domain. For the ring Z of integers, the set of prime elements equals the set of irreducible elements; it's {...−11, −7, −5, −3, −2, 2, 3, 5, 7, 11, ...}. As an example, we consider the Gaussian integers Z[i], that is, complex numbers o ...

See also:

Prime number, Prime number - Representing natural numbers as products of primes, Prime number - How many prime numbers are there?, Prime number - Finding prime numbers, Prime number - Some properties of primes, Prime number - Open questions, Prime number - The largest known prime, Prime number - Applications, Prime number - Primality tests, Prime number - Some special types of primes, Prime number - Prime gaps, Prime number - Formulae yielding prime numbers, Prime number - Generalizations, Prime number - Prime elements in rings, Prime number - Prime ideals, Prime number - Primes in valuation theory, Prime number - Quotes, Prime number - Primes in pop culture

Read more here: » Prime number: Encyclopedia II - Prime number - Generalizations

Main article formula for primes There is no formula for primes which is more efficient at finding primes than the methods mentioned above under "Finding prime numbers". Those which do exist have little practical value. The curious polynomial f(n) = n2 âˆ’ n + 41 yields primes for n = 0,..., 40, but f(41) is composite. It has been proved that there i ...

See also:

Prime number, Prime number - Representing natural numbers as products of primes, Prime number - How many prime numbers are there?, Prime number - Finding prime numbers, Prime number - Some properties of primes, Prime number - Open questions, Prime number - The largest known prime, Prime number - Applications, Prime number - Primality tests, Prime number - Some special types of primes, Prime number - Prime gaps, Prime number - Formulae yielding prime numbers, Prime number - Generalizations, Prime number - Prime elements in rings, Prime number - Prime ideals, Prime number - Primes in valuation theory, Prime number - Quotes, Prime number - Primes in pop culture

Read more here: » Prime number: Encyclopedia II - Prime number - Formulae yielding prime numbers

Let pn denote the n-th prime number (i.e. p1 = 2, p2 = 3, etc.). The gap gn between the consecutive primes pn and pn + 1 is the number of (composite) numbers between them, i.e. gn = pn + 1 ∠...

See also:

Prime number, Prime number - Representing natural numbers as products of primes, Prime number - How many prime numbers are there?, Prime number - Finding prime numbers, Prime number - Some properties of primes, Prime number - Open questions, Prime number - The largest known prime, Prime number - Applications, Prime number - Primality tests, Prime number - Some special types of primes, Prime number - Prime gaps, Prime number - Formulae yielding prime numbers, Prime number - Generalizations, Prime number - Prime elements in rings, Prime number - Prime ideals, Prime number - Primes in valuation theory, Prime number - Quotes, Prime number - Primes in pop culture

Read more here: » Prime number: Encyclopedia II - Prime number - Prime gaps

A scheme X is a locally ringed space with a covering by open sets Ui, such that the restriction of the structure sheaf OX to each Ui gives a locally ringed space of type Spec(Ai) (where Ai is some commutative ring), up to isomorphism of locally ringed spaces. In the early days, this was called a prescheme, and a scheme was defined to be a separated prescheme. The term prescheme has fallen out of use, but can still be found in older books, such as Grothendieck's Éléments de géo ...

See also:

Scheme mathematics, Scheme mathematics - History and motivation, Scheme mathematics - Definitions, Scheme mathematics - The category of schemes, Scheme mathematics - Types of schemes, Scheme mathematics - O_X modules

Read more here: » Scheme mathematics: Encyclopedia II - Scheme mathematics - Definitions

A prime p is called primorial or prime-factorial if it has the form p = Π(n) ± 1 for some number n, where Π(n) stands for the product 2 · 3 · 5 · 7 · 11 · ... of all the primes ≤ n. A prime is called factorial if it is of the form n! ± 1. The first factorial primes are: n! âˆ’ 1 is prime for n = 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166,... (sequence A002982 in OEIS) n! + 1 is prime for n = 1, 2, 3, 11, 27, ...

See also:

Prime number, Prime number - Representing natural numbers as products of primes, Prime number - How many prime numbers are there?, Prime number - Finding prime numbers, Prime number - Some properties of primes, Prime number - Open questions, Prime number - The largest known prime, Prime number - Applications, Prime number - Primality tests, Prime number - Some special types of primes, Prime number - Prime gaps, Prime number - Formulae yielding prime numbers, Prime number - Generalizations, Prime number - Prime elements in rings, Prime number - Prime ideals, Prime number - Primes in valuation theory, Prime number - Quotes, Prime number - Primes in pop culture

Read more here: » Prime number: Encyclopedia II - Prime number - Some special types of primes

Schemes form a category if we take as morphisms the morphisms of locally ringed spaces. Morphisms from schemes to affine schemes are completely understood in terms of ring homomorphisms by the following contravariant adjoint pair: For every scheme X and every commutative ring A we have a natural equivalence Since Z is an initial object in the category of ring ...

See also:

Scheme mathematics, Scheme mathematics - History and motivation, Scheme mathematics - Definitions, Scheme mathematics - The category of schemes, Scheme mathematics - Types of schemes, Scheme mathematics - O_X modules

Read more here: » Scheme mathematics: Encyclopedia II - Scheme mathematics - The category of schemes