In algebra, Zariski's finiteness theorem gives a positive answer to Hilbert's 14th problem for the polynomial ring in two variables, as a special case...

In algebraic geometry, Zariski's main theorem, proved by Oscar Zariski (1943), is a statement about the structure of birational morphisms stating roughly...

is finitely generated over k. Zariski's formulation was shown to be equivalent to the original problem, for X normal. (See also: Zariski's finiteness theorem...

In algebra, Zariski's lemma, proved by Oscar Zariski (1947), states that, if a field K is finitely generated as an associative algebra over another field...

(1983). "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" [Finiteness theorems for abelian varieties over number fields]. Inventiones Mathematicae...

Hilbert's irreducibility theorem, conceived by David Hilbert in 1892, states that every finite set of irreducible polynomials in a finite number of variables...

Zariski ring Zariski tangent space Zariski surface Zariski topology Zariski–Riemann surface Zariski space (disambiguation) Zariski's lemma Zariski's main...

of the ideal. Zariski's lemma asserts that if a field is finitely generated as an associative algebra over a field K, then it is a finite field extension...

and closed graph theorems hold Zariski's main theorem – Theorem of algebraic geometry and commutative algebra "The closed graph theorem in various categories"...

In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different...

213, 1861). Often a more general theorem due to Emmy Noether (1933) is given the name, stating that if L/K is a finite Galois extension of fields with...

homeomorphic. One of the first widely recognized metrization theorems was Urysohn's metrization theorem. This states that every Hausdorff second-countable regular...

Tsen's theorem (algebraic geometry) Weber's theorem (algebraic curves) Zariski's connectedness theorem (algebraic geometry) Zariski's main theorem (algebraic...

in terms of special principal rings and principal ideal domains. Zariski–Samuel theorem: Let R be a principal ring. Then R can be written as a direct product...

Bézout's theorem is a statement concerning the number of common zeros of n polynomials in n indeterminates. In its original form the theorem states that...

2-dimensional schemes (including all arithmetic surfaces) by Lipman (1978). Zariski's method of resolution of singularities for surfaces is to repeatedly alternate...

(2001) [1994], "Finiteness theorems", Encyclopedia of Mathematics, EMS Press Grauert, Hans; Remmert, Reinhold (2004). "The Finiteness Theorem". Theory of...

{\displaystyle E} is a finitely generated algebra over F {\displaystyle F} then the field extension is finite. This is called Zariski's lemma. See also integral...

decomposition itself is not difficult. See below. But, by Zariski's connectedness theorem, the last part in the above says that the fiber f ′ − 1 ( s...

compactness theorem. As a corollary (i.e., its contrapositive), the compactness theorem says that every unsatisfiable first-order theory has a finite unsatisfiable...

transform and a Plancherel theorem for unimodular separable locally compact groups of type I and a decomposition theorem for arbitrary representations...

bundles or divisors on X. A salient feature of projective varieties are the finiteness constraints on sheaf cohomology. For smooth projective varieties, Serre...

V\times \mathbb {P} _{k}^{n}\to V} sends Zariski-closed subsets to Zariski-closed subsets. The main theorem of elimination theory is a corollary and a...

whose underlying variety is a projective variety. Chevalley's structure theorem states that every algebraic group can be constructed from groups in those...

finite subcover. X has a sub-base such that every cover of the space, by members of the sub-base, has a finite subcover (Alexander's sub-base theorem)...

finitely many fibers of f {\displaystyle f} are geometrically integral and all fibers are geometrically connected (by Zariski's connectedness theorem)...

automated theorem proving and formal verification of software. Logical formulas are discrete structures, as are proofs, which form finite trees or, more...

theorem. It was proved in the 1950s in independent works by the mathematicians Emil Artin and David Rees; a special case was known to Oscar Zariski prior...

/ / G a {\displaystyle X/\!/\mathbb {G} _{a}} is affine. By Zariski's finiteness theorem, it is true if dim ⁡ X ≤ 3 {\displaystyle \dim X\leq 3} . On...

characterization of quasi-finiteness in terms of stalks. For a general morphism f : X → Y and a point x in X, f is said to be quasi-finite at x if there exist...