In mathematics, a field F is called quasi-algebraically closed (or C1) if every non-constant homogeneous polynomial P over F has a non-trivial zero provided...

In mathematics, a quasi-projective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some...

conjecture that finite fields are quasi-algebraically closed fields (Artin 1982, page x). Let F {\displaystyle \mathbb {F} } be a finite field and { f j } j =...

Br(k) is trivial if k is a finite field or an algebraically closed field (more generally quasi-algebraically closed field; cf. Tsen's theorem). Br ⁡ ( R...

example, in Chapter 1 of Hartshorne a variety over an algebraically closed field is defined to be a quasi-projective variety,: 15  but from Chapter 2 onwards...

the set of all continuously differentiable functions C1 field, a quasi-algebraically closed field C1, the first of four pure modules taken in the Advanced...

abelian category, and so they are closed under operations such as taking kernels, images, and cokernels. The quasi-coherent sheaves are a generalization...

mathematics, a quasi-finite field is a generalisation of a finite field. Standard local class field theory usually deals with complete valued fields whose residue...

over algebraically closed fields. classifying stack An analog of a classifying space for torsors in algebraic geometry; see classifying stack. closed Closed...

its center (an algebraic torus) with a semisimple group. The latter are classified over algebraically closed fields via their Lie algebras. The classification...

In commutative algebra, a quasi-excellent ring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is called...

degree) and pseudo algebraically closed (every absolutely irreducible variety over F has a point defined over F). Every hyperfinite field is pseudo-finite...

k. The separable closure of k is algebraically closed. Every reduced commutative k-algebra A is a separable algebra; i.e., A ⊗ k F {\displaystyle A\otimes...

affine space. More formally, an affine algebraic set is the set of the common zeros over an algebraically closed field k of some family of polynomials in...

interested in the integer solutions. Algebraic geometry is the study of the solutions in an algebraically closed field of multivariate polynomial equations...

Field extension Algebraic extension Splitting field Algebraically closed field Algebraic element Algebraic closure Separable extension Separable polynomial...

quasi-regular semiring, closed semiring, or occasionally a Lehmann semiring (the latter honoring the paper of Daniel J. Lehmann.) Examples of quasi-regular...

non-trivial zero whenever N > dk. Algebraically closed fields are of Diophantine dimension 0; quasi-algebraically closed fields of dimension 1. Discriminant...

induces an isomorphism from the Lie algebra of U to U itself. Unipotent groups over an algebraically closed field of any given dimension can in principle...

that we are working over a fixed, algebraically closed field k (in classical algebraic geometry, k is usually the field of complex numbers). First, we define...

theorem states that a function field K of an algebraic curve over an algebraically closed field is quasi-algebraically closed (i.e., C1). This implies that...

varieties but fields do not. Besides identities, universal algebra is also interested in structural features associated with quasi-identities. A quasi-identity...

and algebra. This relationship is the basis of algebraic geometry. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields...

conjecture that finite fields are quasi-algebraically closed; see Chevalley–Warning theorem The (now disproved) conjecture that any algebraic form over the p-adics...

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

Hilbert's Nullstellensatz suggests an approach to algebraic geometry over any algebraically closed field k : the maximal ideals in the polynomial ring k[x1...

classification of reductive groups is the same over any algebraically closed field. In particular, the simple algebraic groups are classified by Dynkin diagrams, as...

division algebra over a field K is K itself, so that the Brauer group Br(K) is trivial: K is an algebraically closed field. K is a finite field (Wedderburn's...

underlying field, while the extensions require characteristic 0. Let X be a smooth quasi-projective variety over an algebraically closed field, embedded...

and a semisimple algebra over an algebraically closed field. The universal enveloping algebra of a Lie algebra is an associative algebra that can be used...