commutative algebra

Before giving a rigorous definition of a jet, it is useful to examine some special cases. Jet mathematics - Example: One-dimensional case. Suppose that is a real-valued function having at least k+1 derivatives in a neighborhood U of the point x0. Then by Taylor's theorem, where Then the k-jet of f at the point 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

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Let R be any ring, not necessarily commutative. The Jacobson radical of R is the intersection of the annihilators of all simple right R-modules. There are several equivalent characterizations of the Jacobson radical, such as: J(R) is the intersection of the regular maximal right (or left) ideals of R. J(R) is the intersection of all the right (or left) primitive ideals of R. J(R) is the maximal right (or left) quasi-regular right (resp ...

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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

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Consider the set of all nilpotent elements of R, which will be called the nilradical of R (and will be denoted by N(R)). As can be easily seen, the nilradical of R is just the radical of the zero ideal (0). This brings about an alternative definition for the (general) radical of an ideal I in R. Define Rad(I) as the preimage of N(R/I), the nilradical of See 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

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The radical of an ideal I in a commutative ring R, denoted by Rad(I) or √I, is defined as Intuitively, one can think of the radical of I as obtained by taking all the possible roots of elements of I. Rad(I) turns out to be an ideal itself, containing I. An ideal that is equal to its radical is called a radical ideal or said to be radical. ...

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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

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In the 1950s and 1960s, a fruitful collaboration between Serre and the two years younger Alexander Grothendieck led to important foundational work, much of it motivated by the Weil conjectures. Two major foundational papers by Serre were FAC (Faisceaux Algébriques Cohérents, on coherent cohomology) and GAGA. Serre had early on perceived a need to construct more general and refined cohomology theories to tackle these conjectures. In simple terms, the cohomology of a coherent sheaf over a finite field couldn't capture as much t ...

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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

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From 1959 onwards his interests turned towards number theory, in particular class field theory and the theory of complex multiplication. Amongst his most original contributions were: the concept of algebraic K-theory; the Galois representation theory for l-adic cohomology and the conceptions that these representations were 'large'; and the Serre conjecture on mod p representations that made Fermat's last theorem a c ...

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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

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A smooth assignment of a vector at each point of a manifold is called a vector field. Specifically, a vector field on a manifold M is a smooth map such that the image of x, denoted Vx, lies in Tx(M), the tangent space to x. In the language of fiber bundles, such a map is called a section. A vector field on M is theref ...

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

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The simplest example is that of Rn. In this case the tangent bundle is trivial and isomorphic to R2n. Another simple example is the unit circle, S1. The tangent bundle is of the circle is also trivial and isomorphic to S1 × R. Geometrically, this is a cylinder of infinite height. Unfortunately, the only tangent bundles that can be readily visualized are those of the real line R and the unit circle S1, both of which are trivial. For 2-dimensional manifolds the tangent bundle is 4-dim ...

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

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A polynomial function in one real variable can be represented by a graph. The graph of the zero polynomial f(x) = 0 is the x-axis. The graph of a degree 0 polynomial f(x) = a0 , where a0 ≠ 0, is a horizontal line with y-intercept a0 The graph of a degree 1 polynomial (or linear function) f(x) = a0 + a1x , whe ...

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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

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This subsection deals with the notion of jets of local sections a vector bundle. Almost everything in this section generalizes mutatis mutandis to the case of local sections of a fibre bundle, a Banach bundle over a Banach manifold, a fibered manifold, or quasi-coherent sheaves over schemes. Furthermore, these examples of possible generalizations are certainly not exhaustive. Suppose that E is a finite-dimensional smooth vector bundle over a manifold M, with projection . Then sections of E are smooth functions su ...

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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

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We can generalize the definition of a graded algebra to an arbitrary monoid G as an index set. A G-graded algebra A is an algebra with a direct sum decomposition such that A graded algebra is then the same thing as a N-graded algebra, where N is the monoid of natural numbers. (If we don't require that the ring has an identity element, we ca ...

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Graded algebra, Graded algebra - Graded algebra, Graded algebra - G-graded algebra, Graded algebra - Graded modules

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A Gröbner basis is characterised by any one of the following properties, stated relative to some monomial order: the ideal given by the leading terms of polynomials in the ideal I is itself generated by the leading terms of the basis G; the leading term of any polynomial in I is divisible by the leading term of some polynomial in the basis G; multivariate division of any polynomial in the polynomial ring R by G gives a unique remainder; multivariate division of any polynomial in t ...

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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

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The elementary symmetric polynomials appear when we expand That is, when we substitute values for the variables , we obtain the univariate polynomial whose roots are those values by plugging them into the elementary symmetric polynomials. In the case of the characteristic polynomial of a linear operator, the roots are the eigenvalues of the operator. When we plug these eigenvalues into the elementary symmetric polynomials, we obtain certain numerical invariants of the operator (namely, the coefficients of t ...

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Elementary symmetric polynomial, Elementary symmetric polynomial - Definition, Elementary symmetric polynomial - Properties

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The corresponding idea in module theory is that of a graded module, namely a module M over A such that also and This idea is much used in commutative algebra, and elsewhere, to define under mild hypotheses a Hilbert function, namely the length of Mn as a function of n. Again under mild hypotheses of finiteness, this function is a polynomial, the Hilbert polynomial, for all large enough values ...

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Graded algebra, Graded algebra - Graded algebra, Graded algebra - G-graded algebra, Graded algebra - Graded modules

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Of the cleanly-formulated Hilbert problems, problems 3, 7, 10, 11, 13, 14, 17, 19 and 20 have a resolution that is accepted by consensus. On the other hand, problems 1, 2, 5, 9, 15, 18+, 21, and 22 have solutions that have partial acceptance, but where there exists some controversy as to whether it resolves the problem. The + on 18 denotes that the Kepler problem solution is a computer-assisted proof, a notion anachronistic for a Hilbert problem and also to some extent controversial because of its lack of verifia ...

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Hilbert's problems, Hilbert's problems - Nature and influence of the problems, Hilbert's problems - The problems as Hilbert's manifesto, Hilbert's problems - A round two dozen, Hilbert's problems - Summary, Hilbert's problems - Tabulated information, Hilbert's problems - Footnotes

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This subsection focuses on two different rigorous definitions of the jet of a function at a point, followed by a discussion of Taylor's theorem. These definitions shall prove to be useful later on during the intrinsic definition of the jet of a function between two manifolds. Jet mathematics - An analytic definition. The following definition uses ideas from mathematical analysis to define jets and jet spaces. It can be generalized to smooth functions between Banach spaces, analytic fu ...

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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

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Main article primality test A primality test algorithm is an algorithm which tests a number for primality, i.e. whether the number is a prime number. AKS primality test Fermat primality test Lucas-Lehmer test Lucas-Lehmer primality test Solovay-Strassen primality test Miller-Rabin primality test A probable prime is an integer which, by virtue of having passed a certain test, is considered to be probably prime. Probable primes which are in fact composite (such ...

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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

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There are many ways one can qualify a scheme. According to a basic idea of Grothendieck, conditions should be applied to a morphism of schemes. Any scheme S has a unique morphism to Spec(Z), so this attitude, part of the relative point of view, doesn't lose anything. For detail on the development of scheme theory, which quickly becomes technically demanding, see first glossary of scheme theory. ...

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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

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A root or zero of a polynomial f is a number ζ so that f(ζ) = 0. The fundamental theorem of algebra states that a polynomial of degree n over the complex numbers has exactly n complex roots (not necessarily distinct ones). Therefore a polynomial can be factorized as where each ζi i ...

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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

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Let f(x) be a real-valued function of a real variable. Then f is even if the following equation holds for all real x: f(−x) = f(x) Geometrically, an even function is symmetric with respect to the y-axis, meaning that its graph remains unchanged after reflection about the y-axis. The designation even is not due to the fact that the Taylor series of an even function includes only even powers. ...

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Even and odd functions, Even and odd functions - Even functions, Even and odd functions - Odd functions, Even and odd functions - Some facts, Even and odd functions - Basic properties, Even and odd functions - Series, Even and odd functions - Algebraic structure

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In differential geometry and algebraic geometry it is often convenient to have a coordinate-independent description of differential operators between two vector bundles. Let E and F be two vector bundles over a manifold M. An operator is a mapping of sections, P: Γ(E) → Γ(F) which maps the stalk of the sheaf of germs of Γ(E) at a point x ∈ M to the fibre of F at x: ...

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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

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Differentiation is linear, i.e., D(f + g) = (Df) + (Dg) D(af) = a(Df) where f and g are functions, and a is a constant. Any polynomial in D with function coefficients is also a differential operator. We may also compose differential operators by the rule (D1oDSee 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

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