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Longer titles found: Quasi-algebraically closed field (view), Pseudo algebraically closed field (view)

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an algebraically closed field). Equivalently, a Lie algebra is toral if it contains no nonzero nilpotent elements. Over an algebraically closed field, every

linear algebraic group G{\displaystyle G} over a (not necessarily algebraically closed) field k{\displaystyle k} is the centralizer of a maximal torus. Cartan

the coefficients of the defining equations may not belong to an algebraically closed field. Absolutely irreducible is also applied, with the same meaning

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

linear operator on a finite-dimensional vector space over an algebraically closed field is semisimple if and only if it is diagonalizable. Over a perfect

p{\displaystyle {\mathfrak {p}}}. These conditions are equivalent over an algebraically closed field of characteristic zero, such as the complex numbers. If the field

function field of the affine curve. A smooth connected curve over an algebraically closed field is called hyperbolic if 2g−2+r>0{\displaystyle 2g-2+r>0} where

connected and solvable linear algebraic group defined over an algebraically closed field and ρ : G → G L ( V ) {\displaystyle \rho \colon G\to GL(V)} a

contrast to solvable Lie algebras. Semisimple Lie algebras over an algebraically closed field of characteristic zero are completely classified by their root

flat family of nonsingular varieties. If S is the spectrum of an algebraically closed field and f is of finite type, then one recovers the definition of a

the theory of Lie algebras, Lie's theorem states that, over an algebraically closed field of characteristic zero, if π:g→gl(V){\displaystyle \pi :{\mathfrak

group G acting transitively on an algebraic variety X over an algebraically closed field k and Vi→X,i=1,2{\displaystyle V_{i}\to X,i=1,2} morphisms of

being isogenous is an equivalence relation between tori. Over any algebraically closed field k=k¯{\displaystyle k={\overline {k}}} there is up to isomorphism

henselization of k[T]{\displaystyle k[T]}, k{\displaystyle k} an algebraically closed field, if X{\displaystyle X} is essentially smooth over k{\displaystyle

affine varieties. Let X be an affine algebraic variety over an algebraically closed field k of characteristic zero with the action of a reductive algebraic

{\displaystyle k} ) and showed with Stéphane Lamy that for an algebraically closed field k {\displaystyle k} and for dimension n {\displaystyle n} =2 the

that almost all fibers are smooth curves of genus 1. (Over an algebraically closed field such as the complex numbers, these fibers are elliptic curves

over a prime field with different numbers of points. Over an algebraically closed field of characteristic 2 there is (up to isomorphism) exactly one supersingular

stereographic projection. In particular, every quadric over an algebraically closed field is rational. A quadric over a field k is called isotropic if it

cubic curve is known to have nine points of inflection, over an algebraically closed field such as the complex numbers. This can be shown by taking the homogeneous

Kollár–Miyaoka–Mori showed that a smooth Fano variety over an algebraically closed field is rationally chain connected; that is, any two closed points

Manin (1963). It describes the structure of F-isocrystals over an algebraically closed field k. The category of such F-isocrystals is abelian and semisimple

language of rings is true in some (or equivalently, in every) algebraically closed field of characteristic 0 (such as the complex numbers for instance)

with Haboush's theorem show that if a reductive group (over an algebraically closed field) acts on a finitely generated algebra then the fixed subalgebra

conjecture would imply that every supersingular K3 surface over an algebraically closed field has Picard number 22. This is now known in every characteristic

generally, the fundamental group of a proper scheme over any algebraically closed field of characteristic zero is known, because an extension of algebraically

over k. Every variety over a perfect field (in particular an algebraically closed field) is generically smooth. Affine space and projective space are

K(C) of a nonsingular integral algebraic curve C defined over an algebraically closed field K of characteristic p. The question addresses the existence of

non-negative integer. In the special case when Y is the spectrum of an algebraically closed field (a point), Rqf∗(F ) is the same as Hq(F ). Suppose that X is a

i/n}T^{1/n}.} This construction works if C is replaced by any algebraically closed field C of characteristic zero. (Artin & Tate 2009, §XI.3) say that

The category of coherent sheaves on a complete variety over an algebraically-closed field. The category of finitely-generated projective modules over the

algebraically closed field F of characteristic zero. Moreover, any fixed statement φ holds in C if and only if it holds in any algebraically closed field

finitely generated over k is regular over k. Any extension of an algebraically closed field is regular. An extension is regular if and only if it is separable

over one-dimensional bases. Let X be a normal variety over an algebraically closed field k¯{\displaystyle {\bar {k}}} and U⊂X{\displaystyle U\subset X}

V{\displaystyle V} is a finite-dimensional vector space over an algebraically closed field of characteristic zero, and g{\displaystyle {\mathfrak {g}}} is

theorem states that for any complete varieties U, V and W over an algebraically closed field, and given points u, v and w on them, any invertible sheaf L which

inverse limit of ℓn roots of unity). For abelian varieties over an algebraically closed field K, the Weil pairing is a nondegenerate pairing A[n]×A∨[n]⟶μn{\displaystyle

categorical theory is either locally modular or interprets an algebraically closed field. Cherlin's Question. Is there a maximal (with respect to expansions)

true. However, its opposite poset has deviation 1. Let k be an algebraically closed field and consider the poset of ideals of the polynomial ring k[x] in

indecomposable matrices classify the affine Lie algebras (say over some algebraically closed field of characteristic 0). The determinants of the Cartan matrices

formal Laurent series in T−1 will be useful. It is a complete and algebraically closed field. First we need analogues to the factorials, which appear in the

graph encoding the structure of a reductive group over a non-algebraically-closed field, in which vertices are colored black or white according to whether

theorem shows that the real numbers have Rolle's property. Any algebraically closed field such as the complex numbers has Rolle's property. However, the

sum of a split space and an anisotropic space.: 56  If F is an algebraically closed field, for example, the field of complex numbers, and (V, q) is a quadratic

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

coroots. If G is a connected reductive algebraic group over the algebraically closed field K, then its Langlands dual group LG is the complex connected reductive

{\displaystyle k} . If A {\displaystyle \mathbf {A} } is a matrix over an algebraically closed field in Jordan normal form with respect to some basis, then A {\displaystyle

dimension of A. Given a system of polynomial equations over an algebraically closed field K{\displaystyle K}, it may be difficult to compute the dimension

be a finite morphism between smooth connected curves over an algebraically closed field, P a closed point of X and Q=f(P){\displaystyle Q=f(P)}. We then

function field. The Riemann–Zariski space of a curve over an algebraically closed field k with function field K is the same as the nonsingular projective

uniruled variety) V is a projective algebraic variety over an algebraically closed field such that through every two points there passes the image of a

is replaced by a one-dimensional function field. Let k be an algebraically closed field of characteristic 0, let C/k be a smooth projective curve of genus

algebraic surface: If S{\displaystyle S} is a smooth surface over an algebraically closed field of characteristic 0{\displaystyle 0}, then the generating function

fixed points, and G is a connected affine algebraic group over an algebraically closed field, then the Lang map is surjective. Define: fa:G→G,fa(x)=x−1aσ(x)

for a finite-dimensional semisimple representation V over an algebraically closed field, the numbers of simple representations up to isomorphisms appearing

conditions does not behave well. Suppose X is the affine line over an algebraically closed field k. For each closed point x of X we can consider the local ring

always rational. If S{\displaystyle S} is a smooth surface over an algebraically closed field of characteristic 0{\displaystyle 0}, then the generating function

the statement applies for smooth and proper varieties over an algebraically closed field. This variant of the Atiyah–Bott fixed point formula was proved

proved that a del Pezzo surface has degree d at most 9. Over an algebraically closed field, every del Pezzo surface is either a product of two projective

to describe some basic uses of projective spaces. Let k be an algebraically closed field, and V be a finite-dimensional vector space over k. The symmetric

example a field of characteristic zero, or a finite field, or an algebraically closed field – then every extension of K is separable so that separable K-algebras

\mathbb {P} _{k}^{n}} of pure dimension r{\displaystyle r} over an algebraically closed field k{\displaystyle k}, there is a pairing ExtOXp(OX,F)×ExtOXr−p(F

which is called p-adic Hodge theory. For a variety X over an algebraically closed field of characteristic p > 0, the ℓ{\displaystyle \ell }-adic cohomology

(sometimes also known as “untwisted”) adjoint form of E7. Over an algebraically closed field, this and its double cover are the only forms; however, over other

generates the whole extension. Purely transcendental extensions of an algebraically closed field occur as function fields of rational varieties. The problem of

theory of the symmetric group.) The formula is also true over an algebraically closed field of characteristic not dividing the order of G. David A. Cox, John

field K (typically the field of rationals) and roots xi in an algebraically closed field extension. Substituting the Xi by the xi and the coefficients

2, every Fano variety (known as a Del Pezzo surface) over an algebraically closed field is rational. A major discovery in the 1970s was that starting

matrices M2(F){\displaystyle M_{2}(F)}; a quaternion algebra over an algebraically closed field is always split. If σ{\displaystyle \sigma } is an embedding of

is developed on a smooth algebraic variety X defined over an algebraically closed field K of characteristic zero, such as K = C. The sheaf of differential

by ψ(T) = T+τ, where A is Fp[T] and L is a suitable complete algebraically closed field containing A. It was described by L. Carlitz in 1935, many years

also called a linearization. Let X be a complete variety over an algebraically closed field acted by a connected reductive group G and L an invertible sheaf

For fixed dimension N, it is true if it holds for at least one algebraically closed field of characteristic 0. Let k[X] denote the polynomial ring k[X1

Non algebraically closed field whose extension by sqrt(–1) is algebraically closed

Lie algebra of n × n {\displaystyle n\times n} matrices over an algebraically closed field k {\displaystyle \mathbb {k} } (such as the complex numbers),

nonsingular algebraic curve C{\displaystyle C} defined over an algebraically closed field k{\displaystyle k} of characteristic p≥0{\displaystyle p\geq 0}

y_{j}\}=x_{i}y_{j}+(-1)^{ij}y_{j}x_{i}\ .} Jordan simple superalgebras over an algebraically closed field of characteristic 0 were classified by Kac (1977). They include

surjective map of the corresponding groups with values in some (non algebraically closed) field. There are similar problems with the points of other algebraic

constants (cf. example above) form a special class, and over an algebraically closed field of characteristic zero, the category of finite groups is equivalent

(sometimes also known as "untwisted") adjoint form of E6. Over an algebraically closed field, this and its triple cover are the only forms; however, over other

semisimple Lie algebra g{\displaystyle {\mathfrak {g}}} over an algebraically closed field of characteristic zero. Let b=h⊕n+⊂g{\displaystyle {\mathfrak

for example, this approach taken in (Jantzen 1987). Let k be an algebraically closed field and G a linear algebraic group (that is, affine algebraic group)

universal axioms of ACFp is a substructure of a sufficiently large algebraically closed field M ⊨ A C F 0 {\displaystyle M\models ACF_{0}} , and additionally

_{k}^{n}} and S=Spec(k){\displaystyle S={\text{Spec}}(k)} for an algebraically closed field, then a non-zero section s∈Γ(O(d)){\displaystyle s\in \Gamma ({\mathcal

universal axioms of ACFp is a substructure of a sufficiently large algebraically closed field M ⊨ A C F 0 {\displaystyle M\models ACF_{0}} , and additionally

split (sometimes also known as “untwisted”) form of E8. Over an algebraically closed field, this is the only form; however, over other fields, there are

generalization of a free algebra and a semisimple algebra over an algebraically closed field. The universal enveloping algebra of a Lie algebra is an associative

z) semigroups finite rank associative algebras over a fixed algebraically closed field with zero squared radical and commutative factor over the radical

Manin (1963). It describes the structure of Dieudonné modules over an algebraically closed field k {\displaystyle k} up to "isogeny". More precisely, it classifies

let X {\displaystyle X} be a smooth projective curve over an algebraically closed field k {\displaystyle k} , G {\displaystyle G} a semisimple, split

irreducible g {\displaystyle {\mathfrak {g}}} -module over an algebraically closed field and f : V → V {\displaystyle f:V\to V} is a homomorphism, then

Modern application of infinitesimals Real closed field – Non algebraically closed field whose extension by sqrt(–1) is algebraically closed Real line –

pairwise coprime polynomials which have no multiple root in an algebraically closed field: g = ∏ i = 1 k p i n i . {\displaystyle g=\prod _{i=1}^{k}p_{i}^{n_{i}}

{\displaystyle {\mathfrak {X}}} locally of finite type over an algebraically closed field k {\displaystyle k} whose stabilizers are affine, and x ∈ X (

is flat. Nullstellensatz German for "zero locus theorem". Over algebraically closed field, the weak Nullstellensatz states that the points of affine space

representation theory of the finite groups of Lie type over an algebraically closed field of characteristic 0". Reflections on the International Medical

of eigenvalues generalizes to the Jordan normal form. Over an algebraically closed field, any matrix A has a Jordan normal form and therefore admits a

{4a^{3}}{4a^{3}+27b^{2}}}}. Isomorphic curves have the same j-invariant; over an algebraically closed field, two curves with the same j-invariant are isomorphic. The supersingular

and i is the square root of −1. The surcomplex numbers form an algebraically closed field (except for being a proper class), isomorphic to the algebraic

automorphic automorphic representation Borel–Weil–Bott theorem Over an algebraically closed field of characteristic zero, the Borel–Weil–Bott theorem realizes an