What is the most difficult thing you have had to prove for a

>>8513719
Took putnam today. I'd say those pretty tough. Only solved A1.

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Let [math] n [/math] be an odd integer [math] >2 [/math] and let [math] f(x)\in \mathbb{Q}[x] [/math] be an irreducible polynomial of degree [math] n [/math] such that the Galois group [math] Gal(f/\mathbb{Q}) [/math] is isomorphic to the dihedral group [math] D_n [/math] of order [math] 2n [/math]. Let [math] \alpha [/math] be a real root of [math] f(x) [/math]. Prove [math] \alpha [/math] can be expressed by real radicals if and only if every prime divisor of [math] n [/math] is a Fermat prime.

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Let [math] R [/math] be a ring.. Does [math]R^m \cong R^n [/math] (as left [math] R [/math]-modules) imply [math] m=n[/math]?

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>>8513725
I heard there was a question involving some group theory, do you remember it?

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>>8513771
i feel like this is only true for vector spaces, but can't seem to think of a proof

are you going to post a proof?

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>>8513801
It's true for certain nice rings

Needless to say no one in the class came up with the counter example ring (from Hungerford's algebra book) of infinite matrices with finitely many non-zero elements in each column, which satisfies [math] R\cong R^2 [/math].

(see https://en.wikipedia.org/wiki/Invariant_basis_number)

>>8513719
Construct a linear flow on the 3-torus such that each orbit is dense.
I got close, but I didn't know about Kronecker's theorem.

>>8513719
that sqrt(2) is irrational