; Pseudo algebraically closed field: A field in which every variety has a rational point.

* Ajar, Émile ( Romain Gary ), Hocus Bogus, Yale University Press, 2010, 224p, ISBN 978-0-300-14976-0 ( translation of Pseudo by David Bellos, includes The Life and Death of Émile Ajar )

Pseudo Echo issued a double-CD Teleporter ( 2000 ), produced by Canham.

* Francois Treves, Introduction to Pseudo Differential and Fourier Integral Operators, ( University Series in Mathematics ), Plenum Publ.

Cheyne started working on a solo album with Brian Canham, formerly of Australian synth pop band Pseudo Echo, and Ewen McArthur.

In the 1980s, Pseudo Echo had Australian top 20 hits with " Listening ", " A Beat for You ", " Don't Go ", " Love an Adventure ", " Living in a Dream " and their cover of " Funky Town ", which peaked at No. 1 in 1986.

Pseudo Echo reunited in March 1998, with Canham and Gigliotti joined by Danielson on drums and Tony Featherstone on keyboards ( ex-The Badloves ) to tour.

In late 1999, as Pseudo Echo, Canham and Gigliotti were joined by Martin Dambrosi ( Guitar ) and Ben Grayson on keyboards and released an EP Funkytown Y2K: RMX, with six new remixes of " Funky Town ".

* 1999 Pseudo. com Online Network interview with Richard Metzger of The Disinformation Company

Those two names ( Tambur and Hamamašen ) are included in the History of Taron by Pseudo John Mamikonian in a short passage about a war between the ruler of Tambur, Hamam, and his maternal uncle the Georgian Prince, which resulted in the destruction of the town to be rebuild by Hamam and be named after him namely Hamamshen.

In the novel, " Artificial Intelligence " has been renamed " Pseudo intelligence " ( Hackworth declares the older term to have been " cheeky ", meaning presumptuous ).

It became Pseudo Echo's only top 40 hit in America, although the band has experienced much greater success in their homeland.

Pseudo addiction is a term which has been used to describe patient behaviors that may occur when pain is undertreated.

this is really a deficiency in the scalar field, namely, that it is not algebraically closed.

Let V be a finite-dimensional vector space over an algebraically closed field F, e.g., the field of complex numbers.

From these results, one sees that the study of linear operators on vector spaces over an algebraically closed field is essentially reduced to the study of nilpotent operators.

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.

As an example, the field of real numbers is not algebraically closed, because the polynomial equation x < sup > 2 </ sup > + 1 = 0 has no solution in real numbers, even though all its coefficients ( 1 and 0 ) are real.

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.

Also, no finite field F is algebraically closed, because if a < sub > 1 </ sub >, a < sub > 2 </ sub >, …, a < sub > n </ sub > are the elements of F, then the polynomial ( x − a < sub > 1 </ sub >)( x − a < sub > 2 </ sub >) ··· ( x − a < sub > n </ sub >) + 1

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

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

Given a field F, the assertion “ F is algebraically closed ” is equivalent to other assertions:

The field F is algebraically closed if and only if the only irreducible polynomials in the polynomial ring F are those of degree one.

If F is algebraically closed and p ( x ) is an irreducible polynomial of F, then it has some root a and therefore p ( x ) is a multiple of x − a.

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

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

A field with no proper algebraic extensions is called algebraically closed.

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.

The algebraic closure of K is also the smallest algebraically closed field containing K,

because if M is any algebraically closed field containing K, then the elements of M which are algebraic over K form an algebraic closure of K.

* There are many countable algebraically closed fields within the complex numbers, and strictly containing the field of algebraic numbers ; these are the algebraic closures of transcendental extensions of the rational numbers, e. g. the algebraic closure of Q ( π ).

The central idea of Diophantine geometry is that of a rational point, namely a solution to a polynomial equation or system of simultaneous equations, which is a vector in a prescribed field K, when K is not algebraically closed.

In general, algebraically closed fields are easier to handle than non-algebraically closed ones.

In mathematics, especially in the area of abstract algebra known as module theory, algebraically compact modules, also called pure-injective modules, are modules that have a certain " nice " property which allows the solution of infinite systems of equations in the module by finitary means.

In mathematics, diophantine geometry is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry over a ground field K that is not algebraically closed, such as the field of rational numbers or a finite field, or more general commutative ring such as the integers.

In mathematics, a compactly generated ( topological ) group is a topological group G which is algebraically generated by one of its compact subsets.

In mathematics, a reductive group is an algebraic group G over an algebraically closed field such that the unipotent radical of G is trivial ( i. e., the group of unipotent elements of the radical of G ).

Equivalently ( by definition ), the theorem states that the field of complex numbers is algebraically closed.

For each line or plane of reflection, the symmetry group is isomorphic with Cs ( see point groups in three dimensions ), one of the three types of order two ( involutions ), hence algebraically C2.

Let k be a field ( such as the rational numbers ) and K be an algebraically closed field extension ( such as the complex numbers ), consider the polynomial ring kX < sub > n </ sub > and let I be an ideal in this ring.

Minkowski is perhaps best known for his work in relativity, in which he showed in 1907 that his former student Albert Einstein's special theory of relativity ( 1905 ), presented algebraically by Einstein, could also be understood geometrically as a theory of four-dimensional space-time.

While it is algebraically possible to approximate the effects of anatomy ( the three-equation method ), disease states introduce considerable uncertainty to this approach.

To see the connection with the classical picture, note that for any set S of polynomials ( over an algebraically closed field ), it follows from Hilbert's Nullstellensatz that the points of V ( S ) ( in the old sense ) are exactly the tuples ( a < sub > 1 </ sub >, ..., a < sub > n </ sub >) such that ( x < sub > 1 </ sub >-a < sub > 1 </ sub >, ..., x < sub > n </ sub >-a < sub > n </ sub >) contains S ; moreover, these are maximal ideals and by the " weak " Nullstellensatz, an ideal of any affine coordinate ring is maximal if and only if it is of this form.

The roots of a cubic, like those of a quadratic or quartic ( fourth degree ) function but no higher degree function ( by the Abel – Ruffini theorem ), can always be found algebraically ( as a formula involving simple functions like the square root and cube root functions ).

Every H * is very special in structure: it is pure-injective ( also called algebraically compact ), which says more or less that solving equations in H * is relatively straightforward.

One way that leads to generalisations is to allow reducible algebraic sets ( and fields k that aren't algebraically closed ), so the rings R may not be integral domains.

Over an algebraically closed field, this and its triple cover are the only forms ; however, over other fields, there are often many other forms, or “ twists ” of E < sub > 6 </ sub >, which are classified in the general framework of Galois cohomology ( over a perfect field k ) by the set H < sup > 1 </ sup >( k, Aut ( E < sub > 6 </ sub >)) which, because the Dynkin diagram of E < sub > 6 </ sub > ( see below ) has automorphism group Z / 2Z, maps to H < sup > 1 </ sup >( k, Z / 2Z ) = Hom ( Gal ( k ), Z / 2Z ) with kernel H < sup > 1 </ sup >( k, E < sub > 6, ad </ sub >).

Not all matrices are diagonalizable, but at least over the complex numbers ( or any algebraically closed field ), every matrix is similar to a matrix in Jordan form.

In Weil's main foundational book ( 1946 ), generic points are constructed by taking points in a very large algebraically closed field, called a universal domain.

This can happen in two ways: either it is a reducible variety, meaning that its defining quadratic factors as the product of two linear polynomials ( degree 1 ), or the polynomial is irreducible but does not define a curve, but instead a lower-dimension variety ( a point or the empty set ); this latter can only occur over a field that is not algebraically closed, such as the real numbers.

In the special case when Y is the spectrum of an algebraically closed field ( a point ), R < sup > q </ sup > f < sub >*</ sub >( F ) is the same as

If σ, ..., σ are the distinct K-linear field embeddings of L into an algebraically closed field containing K ( where n is the degree of the extension L / K ), then

The Liouville numbers, denoted L ( not to be confused with Liouville numbers in the sense of rational approximation ), form the smallest algebraically closed subfield of C closed under exponentiation and logarithm ( formally, intersection of all such subfields )— that is, numbers which involve explicit exponentiation and logarithms, but allow explicit and implicit polynomials ( roots of polynomials ); this is defined in.

The distinction is because surgery theory works in dimension 5 and above ( in fact, it works topologically in dimension 4, though this is very involved to prove ), and thus the behavior of manifolds in dimension 5 and above is controlled algebraically by surgery theory.

that the field K is sufficiently large ( for example, K algebraically closed suffices ), otherwise some statements need refinement.

The ' twisted ' nature of objects over fields that are not algebraically closed, which are not isomorphic but become so over the algebraic closure, was also known in many cases linked to other algebraic groups ( such as quadratic forms, simple algebras, Severi – Brauer varieties ), in the 1930s, before the general theory arrived.

Finite type indecomposable matrices classify the finite dimensional simple Lie algebras ( of types ), while affine type indecomposable matrices classify the affine Lie algebras ( say over some algebraically closed field of characteristic 0 ).

Formally, we define a bad field as a structure of the form ( K, T ), where K is an algebraically closed field and T is an infinite, proper, distinguished subgroup of K, such that ( K, T ) is of finite Morley rank in its full language.

In a general case ( if a certain linear function of electromagnetic field does not vanish identically ), three out of four components of the spinor function in the Dirac equation can be algebraically eliminated, yielding an equivalent fourth-order partial differential equation for just one component.