Fundamental Theorems of Mathematics

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Can you name the fundamental theorems of mathematics?

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Theorem SummaryFundamental Theorem of...
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.
Every subgroup of a ______ group is cyclic.
Any integer greater than 1 can be written as a unique product of prime numbers.
The maxima and minima of a linear function over a convex polygonal region occur at the region's corners.
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.
Indefinite integration can be reversed by differentiation.
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.
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.
Every non-constant single-variable polynomial with complex coefficients has at least one complex root.
Every ______ ______ ______ group G is isomorphic to a direct sum of primary cyclic groups and infinite cyclic groups.

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