Science / Fundamental Theorems of Mathematics

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Can you name the fundamental theorems of mathematics?

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Theorem SummaryFundamental Theorem of...
Indefinite integration can be reversed by differentiation.
Every time you play a hand differently from the way you would have played it if you could see all your opponents' cards, they gain.
Any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of a curl-free vector field and a divergence-free vector field.
Every non-constant single-variable polynomial with complex coefficients has at least one complex root.
Given a field extension E /F which is finite and ______, there is a one-to-one correspondence between its intermediate fields and subgroups of its______ group.
The maxima and minima of a linear function over a convex polygonal region occur at the region's corners.
Every ______ ______ ______ group G is isomorphic to a direct sum of primary cyclic groups and infinite cyclic groups.
Every subgroup of a ______ group is cyclic.
Any regular curve with non-zero curvature has its shape completely determined by its curvature and torsion.
Any integer greater than 1 can be written as a unique product of prime numbers.
On any ______ manifold there is a unique torsion-free metric connection.

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