Definition & Betydelse | Engelska ordet ALGEBRAICALLY

Definition av ALGEBRAICALLY

1. algebraiskt

Exempel på hur man kan använda ALGEBRAICALLY i en mening

• The same argument proves that no subfield of the real field is algebraically closed; in particular, the field of rational numbers is not algebraically closed.
• In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.
• The geometric definition of constructible numbers motivates a corresponding definition of constructible points, which can again be described either geometrically or algebraically.
• If the rational root test finds no rational solutions, then the only way to express the solutions algebraically uses cube roots.
• The maximal ideals of R correspond to the points of A (because K is algebraically closed), and the prime ideals of R correspond to the subvarieties of A (an algebraic set is called irreducible or a variety if it cannot be written as the union of two proper algebraic subsets).
• Period (algebraic geometry), numbers that can be expressed as integrals of algebraic differential forms over algebraically defined domains, forming a ring.
• In finite-dimensional systems, the evolving variable may be represented algebraically as an n-dimensional vector.
• The generalization of the Zariski topology to the set of prime ideals of a commutative ring follows from Hilbert's Nullstellensatz, that establishes a bijective correspondence between the points of an affine variety defined over an algebraically closed field and the maximal ideals of the ring of its regular functions.
• An affine variety over an algebraically closed field is conceptually the easiest type of variety to define, which will be done in this section.
• Some abstract problems have been rigorously proved to be unsolvable, such as squaring the circle and trisecting the angle using only the compass and straightedge constructions of classical geometry, and solving the general quintic equation algebraically.
• The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities.
• By maximality, an algebraically independent subset S of L over K is a transcendence basis if and only if L is an algebraic extension of K(S), the field obtained by adjoining the elements of S to K.
• Further, all the maximal algebraically independent subsets have the same cardinality, known as the transcendence degree of the extension.
• ; Frobenius field: A pseudo algebraically closed field whose absolute Galois group has the embedding property.
• Like in the non-graded case, this Hopf algebra can be described purely algebraically as the universal enveloping algebra of the Lie superalgebra.
• The case of an algebraically closed ground field is especially important in algebraic geometry, because, in this case, the homeomorphism above is a map between the affine space and the set of all maximal ideals of the ring of functions (this is Hilbert's Nullstellensatz).
• The composition of the latter operation with the right group action, however, yields a ternary operation , which serves as an affine generalization of group multiplication and which is sufficient to both characterize a principal homogeneous space algebraically and intrinsically characterize the group it is associated with.
• For an algebraically closed field k, a matrix g in GL(n,k) is called semisimple if it is diagonalizable, and unipotent if the matrix g − 1 is nilpotent.
• These algebraically compact modules are analogous to injective modules, where one can extend all module homomorphisms.
• It is possible to solve the quintic equation if general hypergeometric functions are included, although the solution is far too complicated algebraically to be useful.

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