Math Theorems

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Can you name the the following famous mathematical theorems?

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x^n+y^n=z^n has no integer solutions for n>2Number Theory
The fundamental group of a union of path-connected spaces sharing a basepoint is given by the free product of the fundamental groups of the individual spaces.Algebraic Topoloy
ker f -> ker g -> ker h -> coker f -> coker g -> coker h is exactAlgebra
For an integrable function, the integral over a closed interval is equal to the difference of the antiderivtive evaluated at the endpointsCalculus
If every chain has a maximal element, then there exists a maximal element for the posetMath Foundations
The order of a subgroup divides the order of the group in finite groupsAlgebra
Every bounded entire function is constant.Complex Analysis
The product of any number of compact spaces is compact.Topology
Every number has a unique decomposition into prime factorsNumber Theory
Every nonconstant real polynomial has a complex root.Algebra/Complex Analysis
Every map D^n->D^n has a fixed point.Algebraic Topology
A Sphere is equidecomposable with two disjoint spheres.Set-Theoretic Geometry
There is a bijective correspondence between the subfield of a field extension and teh subgroups of the associated Galois group.Algebra
Every contraction between metric spaces has a fixed point.Analysis
If f is an analytic complex function and $/gamma$ is a closed, rectifiable path homologous to 0, then the integral of f over $/gamma$ is 0Complex Analysis
Nonconstant analytic maps map open sets to open sets.Complex Analysis
For three compact compact subspaces of R^3, there is a single plane that bisects all three compact subsets.Algebraic Topology
Every finitely generated abelian group is isomorphic to a unique direct product of cyclic groups.Algebra
Every finite group has at least one p-group for every prime p dividing the order of the groupAlgebra
An infinite collection of first-order sentences has a model iff every finite subset of it has a modelLogic
Every bounded sequence has a convergent subsequence.Analysis
A subset of R^n is compact iff it is closed and boundedAnalysis
For any group homomorphism f:G->H, G/ker f is isomorphic to im f.Algebra
For sets A and B, if there are injective maps A->B and B->A, then A and B have the same cardinality.Set Theory
Every complete metric space is a Baire space, and every locally compact Hausdorff space is a Baire SpaceTopology/Functional Analysis

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