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FINITENESS THEOREM

Finiteness theorem

Topics referred to by the same term

mathematics, there are several finiteness theorems. Ahlfors finiteness theorem Finiteness theorem for a proper morphism Compactness theorem, in mathematical logic

Finiteness theorem

Finiteness_theorem

Ahlfors finiteness theorem

Mathematical theory

the Ahlfors finiteness theorem describes the quotient of the domain of discontinuity by a finitely generated Kleinian group. The theorem was proved by

Ahlfors finiteness theorem

Ahlfors_finiteness_theorem

Classification of finite simple groups

Theorem classifying finite simple groups

classification of finite simple groups (popularly called the enormous theorem) is a result of group theory stating that every finite simple group is either

Classification of finite simple groups

Classification_of_finite_simple_groups

Zariski's finiteness theorem

Theorem in algebra

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

Zariski's finiteness theorem

Zariski's_finiteness_theorem

Sylow theorems

Theorems that help decompose a finite group based on prime factors of its order

mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter

Sylow theorems

Sylow_theorems

Rank-finiteness

Rowell. The rank-finiteness theorem for G-crossed braided fusion categories is a theorem, also due to Jones et al. The rank-finiteness theorem for super-modular

Rank-finiteness

Kruskal's tree theorem

Well-quasi-ordering of finite trees

In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under

Kruskal's tree theorem

Kruskal's_tree_theorem

Riemannian geometry

Branch of differential geometry

is diffeomorphic to a sphere. Cheeger's finiteness theorem. Given constants C, D and V, there are only finitely many (up to diffeomorphism) compact n-dimensional

Riemannian geometry

Riemannian_geometry

Compactness theorem

Theorem in mathematical logic

compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important

Compactness theorem

Compactness_theorem

Nick Katz

American mathematician (born 1943)

sums) with algebro-geometric methods. He introduced the Katz–Lang finiteness theorem. Gauss sums, Kloosterman sums, and monodromy groups. Annals of Mathematics

Nick Katz

Nick_Katz

Faltings' theorem

Curves of genus > 1 over the rationals have only finitely many rational points

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

Faltings' theorem

Faltings'_theorem

Kolmogorov extension theorem

Consistent set of finite-dimensional distributions will define a stochastic process

extension theorem (also known as Kolmogorov existence theorem, the Kolmogorov consistency theorem or the Daniell-Kolmogorov theorem) is a theorem that guarantees

Kolmogorov extension theorem

Kolmogorov_extension_theorem

Hilbert's basis theorem

Polynomial ideals are finitely generated

mathematics, Hilbert's basis theorem asserts that every ideal of a polynomial ring over a field has a finite generating set (a finite basis in Hilbert's terminology)

Hilbert's basis theorem

Hilbert's_basis_theorem

Hilbert's theorem

Topics referred to by the same term

ring is finitely generated Hilbert's finiteness theorem, in invariant theory, stating that the ring of invariants of a reductive group is finitely generated

Hilbert's theorem

Hilbert's_theorem

Spectral theorem

Result about when a matrix can be diagonalized

on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies

Spectral theorem

Spectral_theorem

Katz–Lang finiteness theorem

On kernels of maps between abelianized fundamental groups of schemes and fields

Katz–Lang finiteness theorem, proved by Nick Katz and Serge Lang (1981), states that if X is a smooth, geometrically connected scheme of finite type over

Katz–Lang finiteness theorem

Katz–Lang_finiteness_theorem

Lagrange's theorem (group theory)

Theorem on the orders of subgroups

the mathematical field of group theory, Lagrange's theorem states that if H is a subgroup of any finite group G, then | H | {\displaystyle |H|} is a divisor

Lagrange's theorem (group theory)

Lagrange's_theorem_(group_theory)

Fubini's theorem

Conditions for switching order of integration in calculus

the Fubini and Tonelli theorems are necessarily somewhat technical, as they have to use a hypothesis related to σ-finiteness. Most proofs involve building

Fubini's theorem

Fubini's_theorem

Dilworth's theorem

On chains and antichains in partial orders

in the areas of order theory and combinatorics, Dilworth's theorem states that, in any finite partially ordered set, the maximum size of an antichain of

Dilworth's theorem

Dilworth's_theorem

Poincaré recurrence theorem

Certain dynamical systems will eventually return to (or approximate) their initial state

physics, the Poincaré recurrence theorem states that certain dynamical systems will, after a sufficiently long but finite time, almost certainly return to

Poincaré recurrence theorem

Poincaré_recurrence_theorem

List of theorems

Hironaka theorem (algebraic geometry) Hodge index theorem (algebraic surfaces) Katz–Lang finiteness theorem (number theory) Lefschetz hyperplane theorem (algebraic

List of theorems

List_of_theorems

David Hilbert

German mathematician (1862–1943)

demonstration in 1888 of his famous finiteness theorem. Twenty years earlier, Paul Gordan had demonstrated the theorem of the finiteness of generators for binary

David Hilbert

David_Hilbert

Carathéodory's extension theorem

Theorem extending pre-measures to measures

In measure theory, Carathéodory's extension theorem (named after the mathematician Constantin Carathéodory) states that any pre-measure defined on a given

Carathéodory's extension theorem

Carathéodory's_extension_theorem

Kazhdan–Margulis theorem

Theorem in Lie theory in mathematics

rigidity and finite generation of lattices the Kazhdan-Margulis theorem is an important ingredient in the proof of Wang's finiteness theorem. If G {\displaystyle

Kazhdan–Margulis theorem

Kazhdan–Margulis_theorem

Lars Ahlfors

Finnish mathematician (1907–1996)

in 1996. Ahlfors finiteness theorem Ahlfors function Ahlfors measure conjecture Beurling–Ahlfors transform Schwarz–Ahlfors–Pick theorem Measurable Riemann

Lars Ahlfors

Lars_Ahlfors

Andreotti–Grauert theorem

Theorem

Superiore di Pisa - Classe di Scienze. 27 (4): 933–997. Parshin, A.N. (2001) [1994], "Finiteness theorems", Encyclopedia of Mathematics, EMS Press v t e

Andreotti–Grauert theorem

Andreotti–Grauert_theorem

Monotone convergence theorem

Theorems on the convergence of bounded monotonic sequences

mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the good convergence behaviour of monotonic

Monotone convergence theorem

Monotone_convergence_theorem

Hall's marriage theorem

Result in combinatorics and graph theory

mathematics, Hall's marriage theorem, proved by Philip Hall (1935), is a theorem with two equivalent formulations. In each case, the theorem gives a necessary and

Hall's marriage theorem

Hall's_marriage_theorem

Heine–Borel theorem

Subset of Euclidean space is compact if and only if it is closed and bounded

open cover of S {\displaystyle S} has a finite subcover S {\displaystyle S} is closed and bounded. The theorem is sometimes also called the Borel–Lebesgue

Heine–Borel theorem

Heine–Borel_theorem

Coherent sheaf cohomology

Concept in algebraic geometry

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

Coherent sheaf cohomology

Coherent_sheaf_cohomology

Bolzano–Weierstrass theorem

Bounded sequence in finite-dimensional Euclidean space has a convergent subsequence

Bolzano–Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean

Bolzano–Weierstrass theorem

Bolzano–Weierstrass_theorem

Mordell–Weil theorem

The group of K-rational points of an abelian variety is a finitely-generated abelian group

proof. Certainly the finiteness of this group is a necessary condition for E ( Q ) {\displaystyle E(\mathbb {Q} )} to be finitely generated; and it shows

Mordell–Weil theorem

Mordell–Weil_theorem

Cauchy's theorem (group theory)

Existence of group elements of prime order

In mathematics, specifically group theory, Cauchy's theorem states that if G is a finite group and p is a prime number dividing the order of G (the number

Cauchy's theorem (group theory)

Cauchy's_theorem_(group_theory)

Euler characteristic

Topological invariant in mathematics

finite by Grothendieck's finiteness theorem. This is an instance of the Euler characteristic of a chain complex, where the chain complex is a finite resolution

Euler characteristic

Euler_characteristic

Nyquist–Shannon sampling theorem

Sufficiency theorem for reconstructing signals from samples

signal of finite bandwidth, such that the original signal can be reconstructed exactly from those samples. Strictly speaking, the theorem only applies

Nyquist–Shannon sampling theorem

Nyquist–Shannon_sampling_theorem

Emmy Noether

German mathematician (1882–1935)

Endlichkeitssatz der Invarianten endlicher Gruppen" [The Finiteness Theorem for Invariants of Finite Groups] (PDF), Mathematische Annalen (in German), 77

Emmy Noether

Emmy_Noether

Finitely generated abelian group

Commutative group where every element is the sum of elements from one finite subset

the fundamental theorem of finite abelian groups. The theorem, in both forms, in turn generalizes to the structure theorem for finitely generated modules

Finitely generated abelian group

Finitely_generated_abelian_group

Sylvester–Gallai theorem

Existence of a line through two points

The Sylvester–Gallai theorem in geometry states that every finite set of points in the Euclidean plane has a line that passes through exactly two of the

Sylvester–Gallai theorem

Sylvester–Gallai_theorem

Structure theorem for finitely generated modules over a principal ideal domain

Statement in abstract algebra

structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian

Structure theorem for finitely generated modules over a principal ideal domain

Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain

Primitive element theorem

Field theory theorem

primitive element theorem states that every finite separable field extension is simple, i.e. generated by a single element. This theorem implies in particular

Primitive element theorem

Primitive_element_theorem

Finite set

Finite collection of distinct objects

numerical concept of finiteness.) Ia-finite. For every partition of S {\displaystyle S} into two sets, at least one of the two sets is I-finite. (A set with this

Finite set

Finite_set

Gödel's completeness theorem

Fundamental theorem in mathematical logic

Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability

Gödel's completeness theorem

Gödel's_completeness_theorem

Maschke's theorem

Concerns the decomposition of representations of a finite group into irreducible pieces

Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group

Maschke's theorem

Maschke's_theorem

Abelian group

Commutative group (mathematics)

fundamental theorem of finitely generated abelian groups. The existence of algorithms for Smith normal form shows that the fundamental theorem of finitely generated

Abelian group

Abelian_group

Infinite monkey theorem

Counterintuitive result in probability

the monkey would almost surely type every possible finite text an infinite number of times. The theorem can be generalized to state that any infinite sequence

Infinite monkey theorem

Infinite_monkey_theorem

Ramsey's theorem

Statement in mathematical combinatorics

colour. An extension of this theorem applies to any finite number of colours, rather than just two. More precisely, the theorem states that for any given

Ramsey's theorem

Ramsey's_theorem

Arithmetic geometry

Branch of algebraic geometry

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

Arithmetic geometry

Arithmetic_geometry

Base change theorems

Relate the direct image and the pull-back of sheaves

1016/0022-4049(88)90102-8 Gabber, "Finiteness theorems for étale cohomology of excellent schemes" Grauert, Hans (1960), "Ein Theorem der analytischen Garbentheorie

Base change theorems

Base_change_theorems

Kleinian group

Discrete group of Möbius transformations

generators. The Ahlfors finiteness theorem says that such a group is of finite type. A Kleinian group Γ has finite covolume if H3/Γ has finite volume. Any Kleinian

Kleinian group

Kleinian_group

Finite model theory

Branch of logic

theory that fail for finite structures under finite model theory include the compactness theorem, Gödel's completeness theorem, and the method of ultraproducts

Finite model theory

Finite_model_theory

Mean value theorem

Theorem in mathematics

In calculus and real analysis, the mean value theorem (or Lagrange's mean value theorem) is a theorem about differentiable functions, roughly stating

Mean value theorem

Mean_value_theorem

Central limit theorem

Fundamental theorem in probability theory and statistics

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample

Central limit theorem

Central_limit_theorem

Feit–Thompson theorem

Classification theorem in group theory

In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved in the early 1960s

Feit–Thompson theorem

Feit–Thompson_theorem

List of inventions and discoveries by women

endlicher linearer Gruppen der Charakteristik p" [The finiteness theorem of the invariants of finite linear groups with the characteristic "p"], Nachrichten

List of inventions and discoveries by women

List_of_inventions_and_discoveries_by_women

Rank–nullity theorem

In linear algebra, relation between 3 dimensions

The rank–nullity theorem is a theorem in linear algebra, which asserts: the number of columns of a matrix M is the sum of the rank of M and the nullity

Rank–nullity theorem

Rank–nullity_theorem

Ham sandwich theorem

Theorem that any three objects in space can be simultaneously bisected by a plane

mathematical measure theory, for every positive integer n the ham sandwich theorem states that given n measurable "objects" in n-dimensional Euclidean space

Ham sandwich theorem

Ham_sandwich_theorem

Σ-finite measure

Concept in measure theory

{\displaystyle \sigma } -finite. A different but related notion that should not be confused with σ {\displaystyle \sigma } -finiteness is s-finiteness. Let ( X , A

Σ-finite measure

Σ-finite_measure

Menger's theorem

Theorem in graph theory

In the mathematical discipline of graph theory, Menger's theorem says that in a finite graph, the size of a minimum cut set is equal to the maximum number

Menger's theorem

Menger's_theorem

Min-max theorem

Theorem in functional analysis

the main theorem uses essentially the same idea from the finite-dimensional argument. In the case that the operator is non-Hermitian, the theorem provides

Min-max theorem

Min-max_theorem

Finite volume method

Method for representing and evaluating partial differential equations

surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a

Finite volume method

Finite_volume_method

Optional stopping theorem

Theorem in probability theory

particular, the theorem applies to doubling strategies and illustrates mathematically why such strategies cannot guarantee a profit with finite resources.

Optional stopping theorem

Optional_stopping_theorem

Wedderburn–Artin theorem

Classification of semi-simple rings and algebras

algebra, the Wedderburn–Artin theorem is a classification theorem for semisimple rings and semisimple algebras. The theorem states that a(n Artinian) semisimple

Wedderburn–Artin theorem

Wedderburn–Artin_theorem

Finite group

Mathematical group based upon a finite number of elements

a theorem – the classification of finite simple groups. Inspection of the list of finite simple groups shows that groups of Lie type over a finite field

Finite group

Finite_group

Pythagorean theorem

Relation between sides of a right triangle

In mathematics, the Pythagorean theorem or Pythagoras's theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle

Pythagorean theorem

Pythagorean_theorem

Finite element method

Numerical method for solving physical or engineering problems

Finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical

Finite element method

Finite_element_method

Isomorphism theorems

Group of mathematical theorems

specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship among quotients

Isomorphism theorems

Isomorphism_theorems

Birkhoff's representation theorem

Equivalence of distributive lattices and set families

Birkhoff's theorem (disambiguation). In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive

Birkhoff's representation theorem

Birkhoff's_representation_theorem

Wedderburn's little theorem

Result in algebra

immediately yields a proof of the theorem as follows: let K be a finite field. Since the Herbrand quotient vanishes by finiteness, Br ⁡ ( K ) = H 2 ( K al /

Wedderburn's little theorem

Wedderburn's_little_theorem

Mirsky's theorem

Characterizes the height of any finite partially ordered set

the areas of order theory and combinatorics, Mirsky's theorem characterizes the height of any finite partially ordered set in terms of a partition of the

Mirsky's theorem

Mirsky's_theorem

Algebraic number theory

Branch of number theory

significant number-theory problem formulated by Waring in 1770. As with the finiteness theorem, he used an existence proof that shows there must be solutions for

Algebraic number theory

Algebraic_number_theory

Radon–Nikodym theorem

Expressing a measure as an integral of another

In mathematics, the Radon–Nikodym theorem, named after Johann Radon and Otto M. Nikodym, is a result in measure theory that expresses the relationship

Radon–Nikodym theorem

Radon–Nikodym_theorem

Hahn–Banach theorem

Theorem on extension of bounded linear functionals

In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace

Hahn–Banach theorem

Hahn–Banach_theorem

Hadamard factorization theorem

Statement in complex analysis

of complex analysis, the Hadamard factorization theorem asserts that every entire function with finite order can be represented as a product involving

Hadamard factorization theorem

Hadamard_factorization_theorem

Simple group

Group without normal subgroups other than the trivial group and itself

for finite groups one eventually arrives at uniquely determined simple groups, by the Jordan–Hölder theorem. The complete classification of finite simple

Simple group

Simple_group

Robertson–Seymour theorem

Finiteness of sets of forbidden graph minors

under taking minors can be defined by a finite set of forbidden minors, in the same way that Wagner's theorem characterizes the planar graphs as being

Robertson–Seymour theorem

Robertson–Seymour_theorem

Gödel's incompleteness theorems

Limitative results in mathematical logic

Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories

Gödel's incompleteness theorems

Gödel's_incompleteness_theorems

Euclid's theorem

Infinitely many prime numbers exist

proofs of the theorem. Euclid offered a proof in his work Elements (Book IX, Proposition 20), which is paraphrased here. Consider any finite list of prime

Euclid's theorem

Euclid's_theorem

Primary decomposition

In algebra, expression of an ideal as the intersection of ideals of a specific type

decomposition, of finitely many primary ideals (which are related to, but not quite the same as, powers of prime ideals). The theorem was first proven

Primary decomposition

Primary_decomposition

Picard–Lindelöf theorem

Existence and uniqueness of solutions to initial value problems

known as Picard's existence theorem, the Cauchy–Lipschitz theorem, or the existence and uniqueness theorem. The theorem is named after Émile Picard,

Picard–Lindelöf theorem

Picard–Lindelöf_theorem

Finite field

Algebraic structure

one gets no new finite structures: Wedderburn's little theorem states that all finite division rings are commutative, and hence are finite fields. The Artin–Zorn

Finite field

Finite_field

Inverse function theorem

Theorem in mathematics

In mathematical analysis, the inverse function theorem gives sufficient conditions for a function to have an inverse function. The essential idea is that

Inverse function theorem

Inverse_function_theorem

Zermelo's theorem (game theory)

In board games that cannot end in a draw, one of the two players has a winning strategy

In game theory, Zermelo's theorem is a theorem about finite two-person games of perfect information in which the players move alternately and in which

Zermelo's theorem (game theory)

Zermelo's_theorem_(game_theory)

Cayley's theorem

Representation of groups by permutations

is finite, Sym ⁡ ( G ) {\displaystyle \operatorname {Sym} (G)} is finite too. The proof of Cayley's theorem in this case shows that if G is a finite group

Cayley's theorem

Cayley's_theorem

Mostow rigidity theorem

Theorem in hyperbolic geometry

rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a complete, finite-volume hyperbolic

Mostow rigidity theorem

Mostow_rigidity_theorem

Uniqueness theorem

Index of articles associated with the same name

Thompson uniqueness theorem in finite group theory. Uniqueness theorem for Poisson's equation. Electromagnetism uniqueness theorem for the solution of

Uniqueness theorem

Uniqueness_theorem

Sheaf (mathematics)

Tool to track locally defined data attached to the open sets of a topological space

f^{-1}} . See inverse image functor. Finiteness conditions for module over commutative rings give rise to similar finiteness conditions for sheaves of modules:

Sheaf (mathematics)

Sheaf_(mathematics)

Brouwer fixed-point theorem

Theorem in topology

Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f

Brouwer fixed-point theorem

Brouwer_fixed-point_theorem

Lusin's theorem

Theorem in measure theory

analysis, Lusin's theorem (or Luzin's theorem, named for Nikolai Luzin) or Lusin's criterion states that an almost-everywhere finite function is measurable

Lusin's theorem

Lusin's_theorem

Chevalley–Warning theorem

Certain polynomial equations in enough variables over a finite field have solutions

theory, the Chevalley–Warning theorem implies that certain polynomial equations in sufficiently many variables over a finite field have solutions. It was

Chevalley–Warning theorem

Chevalley–Warning_theorem

Regular language

Formal language that can be expressed using a regular expression

language recognised by a finite automaton. The equivalence of regular expressions and finite automata is known as Kleene's theorem (after American mathematician

Regular language

Regular_language

Myhill–Nerode theorem

Necessary and sufficient condition for a formal language to be regular

However, it does not necessarily have finitely many states. The Myhill–Nerode theorem shows that finiteness is necessary and sufficient for language

Myhill–Nerode theorem

Myhill–Nerode_theorem

Frucht's theorem

On graphs with given symmetry groups

Frucht's theorem is a result in algebraic graph theory, conjectured by Dénes Kőnig in 1936 and proved by Robert Frucht in 1939. It states that every finite group

Frucht's theorem

Frucht's_theorem

Fermat's Last Theorem

17th-century conjecture proved by Andrew Wiles in 1994

In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that there are no positive integers a

Fermat's Last Theorem

Fermat's_Last_Theorem

Lax equivalence theorem

Theorem in numerical analysis

In numerical analysis, the Lax equivalence theorem is a fundamental theorem in the analysis of linear finite difference methods for the numerical solution

Lax equivalence theorem

Lax_equivalence_theorem

Theorem

In mathematics, a statement that has been proven

mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The proof of a theorem is a logical argument that uses

Theorem

Brun's theorem

Theorem that the sum of the reciprocals of the twin primes converges

theory, Brun's theorem states that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a finite value known

Brun's theorem

Brun's_theorem

Perron–Frobenius theorem

Theorem in linear algebra

This theorem has important applications to probability theory (ergodicity of Markov chains); to the theory of dynamical systems (subshifts of finite type);

Perron–Frobenius theorem

Perron–Frobenius_theorem

Finite geometry

Geometric system with a finite number of points

Incidence is containment. If D is finite then it must be a finite field GF(q), since by Wedderburn's little theorem all finite division rings are fields. In

Finite geometry

Finite_geometry

Four color theorem

Planar maps require at most four colors

In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map

Four color theorem

Four_color_theorem

Zariski's main theorem

Theorem of algebraic geometry and commutative algebra

variations that have been called Zariski's main theorem are as follows: A birational morphism with finite fibers to a normal variety is an isomorphism to

Zariski's main theorem

Zariski's_main_theorem

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