Provide the most elegant proof to the following claim you can produce. Suppose and are nonzero complex numbers and for every positive integer it is true that . then and the are a reordering of the .

# Month: May 2016

# Spooky geometry III

Construct a path-connected metric space and a discontinuous function such that is continuous for any continuous path .

# Galois correspondence via group algebras

Suppose is a finite group of automorphisms of a field . Form the ring which is the subring of the ring of endomorphisms of generated by and multiplication by elements of . Note that if then the composite equals the composite . Show that the -subalgebras of the group ring are those of the form for a subgroup of .

*Note* By an -subalgebra we mean those subrings that contain all the multiplications by elements .

# Spooky geometry II

Find a compact metric space that does not embed in for any .

# Compact isometries

As seen inĀ this post, any isometry from a compact metric space to itself is surjective, so the set of all such isometries is actually a group.

Give a reasonable metric and prove that it is compact as well. What can one say about the sequence of iterates ?

# Duality in polynomial rings

Let be a polynomial ring over a commutative ring (!) in variables. Show that for any -module and , there are natural isomorphisms

# Balanced signs

Assume is a nonincreasing sequence of positive integers whose sum diverges and assume signs are chosen so that the modified series converges. Then

Assume now only the signed series converges, with no hypothesis on the unsigned sequence. Show