** Every field has an algebraic closure.

In fact, it is the smallest algebraically closed field containing the rationals, and is therefore called the algebraic closure of the rationals.

Every field has an algebraic extension which is algebraically closed ( called its algebraic closure ), but proving this in general requires some form of the axiom of choice.

In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.

Using Zorn's lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K.

The algebraic closure of a field K can be thought of as the largest algebraic extension of K.

To see this, note that if L is any algebraic extension of K, then the algebraic closure of L is also an algebraic closure of K, and so L is contained within the algebraic closure of K.

Another example of an algebraically closed field is the field of ( complex ) algebraic numbers.

The sum, difference, product and quotient of two algebraic numbers is again algebraic ( this fact can be demonstrated using the resultant ), and the algebraic numbers therefore form a field, sometimes denoted by A ( which may also denote the adele ring ) or < span style =" text-decoration: overline ;"> Q </ span >.

This can be rephrased by saying that the field of algebraic numbers is algebraically closed.

The name algebraic integer comes from the fact that the only rational numbers which are algebraic integers are the integers, and because the algebraic integers in any number field are in many ways analogous to the integers.

If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O < sub > K </ sub >.

Then, following the programme he outlined in his talk at the 1958 International Congress of Mathematicians, he introduced the theory of schemes, developing it in detail in his Éléments de géométrie algébrique ( EGA ) and providing the new more flexible and general foundations for algebraic geometry that has been adopted in the field since that time.

In abstract algebra, a field extension L / K is called algebraic if every element of L is algebraic over K, i. e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i. e. which contain transcendental elements, are called transcendental.

For example, the field extension R / Q, that is the field of real numbers as an extension of the field of rational numbers, is transcendental, while the field extensions C / R and Q (√ 2 )/ Q are algebraic, where C is the field of complex numbers.

For instance, the field of all algebraic numbers is an infinite algebraic extension of the rational numbers.

If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.

* Hence, the set of algebraic numbers has Lebesgue measure zero ( as a subset of the complex numbers ), i. e. " almost all " complex numbers are not algebraic.

* Given an algebraic number, there is a unique monic polynomial ( with rational coefficients ) of least degree that has the number as a root.

If its minimal polynomial has degree, then the algebraic number is said to be of degree.

AES has a fairly simple algebraic description.

Within algebraic geometry itself, his theory of schemes has become the universally accepted language for all further technical work.

His construction of new cohomology theories has left deep consequences for algebraic number theory, algebraic topology, and representation theory.

His conjectural theory of motives has been a driving force behind modern developments in algebraic K-theory, motivic homotopy theory, and motivic integration.

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.

A near-field is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws.

Diophantine geometry, which is the application of techniques from algebraic geometry in this field, has continued to grow as a result ; since treating arbitrary equations is a dead end, attention turns to equations that also have a geometric meaning.

Edward Fredkin in the Fredkin Finite Nature Hypothesis has suggested an informational basis for Einstein's hypothetical algebraic system.

* Linear algebraic groups ( or more generally affine group schemes ) — These are the analogues of Lie groups, but over more general fields than just R or C. Although linear algebraic groups have a classification that is very similar to that of Lie groups, and give rise to the same families of Lie algebras, their representations are rather different ( and much less well understood ).

In this setting, a homomorphism ƒ: A → B is a function between two algebraic structures of the same type such that

They provide a natural framework for analysing the continuous symmetries of differential equations ( differential Galois theory ), in much the same way as permutation groups are used in Galois theory for analysing the discrete symmetries of algebraic equations.

It is straightforward to find algebraic equations the solutions of which have the topology of a Möbius strip, but in general these equations do not describe the same geometric shape that one gets from the twisted paper model described above.

In universal algebra, a subalgebra of an algebra A is a subset S of A that also has the structure of an algebra of the same type when the algebraic operations are restricted to S. If the axioms of a kind of algebraic structure is described by equational laws, as is typically the case in universal algebra, then the only thing that needs to be checked is that S is closed under the operations.

Random vectors can be subjected to the same kinds of algebraic operations as can non-random vectors: addition, subtraction, multiplication by a scalar, and the taking of inner products.

In comparison, abbreviated algebraic notation represents the same moves with fewer characters, on average, and can avoid confusion since it always represents the same square in the same way.

* Video Orbits, 1993: Mann was the first to produce an algorithm for automatically combining multiple pictures of the same subject matter, using algebraic projective geometry, to " stitch together " images using automatically estimated perspective correction.

Universal algebra defines a notion of kernel for homomorphisms between two algebraic structures of the same kind.

A well-known example was the development of analytic geometry, which in the hands of mathematicians such as Descartes and Fermat showed that many theorems about curves and surfaces of special types could be stated in algebraic language ( then new ), each of which could then be proved using the same techniques.

In finite dimensions, every linear functional is continuous, so the continuous dual is the same as the algebraic dual, although this is not true in infinite dimensions.

Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory.

In the case g = 1, the notion of abelian variety is the same as that of elliptic curve, and every complex torus gives rise to such a curve ; for g > 1 it has been known since Riemann that the algebraic variety condition imposes extra constraints on a complex torus.

# Systems in which the two equations represent the same set of points: they are mathematically equivalent ( one equation can typically be transformed into the other through algebraic manipulation ).

There is a triple equivalence of categories between the category of smooth projective algebraic curves over the complex numbers, the category of compact Riemann surfaces, and the category of complex algebraic function fields, so that in studying these subjects we are in a sense studying the same thing.

General problems of so-called ' descent ' in algebraic geometry were considered, at the same period when the fundamental group was generalised to the algebraic geometry setting ( as a pro-finite group ).

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X ( for example X could be a topological space, a manifold, or an algebraic variety ): to every point x of the space X we associate ( or " attach ") a vector space V ( x ) in such a way that these vector spaces fit together to form another space of the same kind as X ( e. g. a topological space, manifold, or algebraic variety ), which is then called a vector bundle over X.