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...

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...

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...

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...

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 =...

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

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

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

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...

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...

finite fields are not algebraically closed, they are quasi-algebraically closed, which means that every homogeneous polynomial over a finite field has a...

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...

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

of classifying the simple Lie algebras. The simple Lie algebras of finite dimension over an algebraically closed field F of characteristic zero were classified...

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

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...

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...

domains that are finitely generated algebras over an algebraically closed field k, then, working with only the closed points, the above coincides with the...

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

whenever N >  dk. Algebraically closed fields are of Diophantine dimension 0; quasi-algebraically closed fields of dimension 1. Clearly if a field is Ti then...

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...

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

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

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

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...

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...

a field, it has n roots, r 1 , r 2 , … , r n {\displaystyle r_{1},r_{2},\dots ,r_{n}} , not necessarily all distinct, in any algebraically closed extension...

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

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