Fundamental Theorems of Mathematics

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Can you name the fundamental theorems of mathematics?

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Theorem SummaryFundamental Theorem of...
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.
Every subgroup of a ______ group is cyclic.
Every ______ ______ ______ group G is isomorphic to a direct sum of primary cyclic groups and infinite cyclic groups.
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 integer greater than 1 can be written as a unique product of prime numbers.
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.
Indefinite integration can be reversed by differentiation.
On any ______ manifold there is a unique torsion-free metric connection.
Any regular curve with non-zero curvature has its shape completely determined by its curvature and torsion.

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