# Sum of powers

Provide the most elegant proof to the following claim you can produce. Suppose $z_1,\ldots,z_s$ and $w_1,\ldots,w_r$ are nonzero complex numbers and for every positive integer $n$ it is true that $z_1^n +\cdots+z_s^n = w_1^n+ \cdots+w_r^n$. then $r=s$ and the $w_i$ are a reordering of the $z_j$.

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# Spooky geometry III

Construct a path-connected metric space $X$ and a discontinuous function $f:X\to \Bbb{R}$ such that $f\circ \sigma$ is continuous for any continuous path $\sigma:[0,1]\to X$.

# Galois correspondence via group algebras

Suppose $G$ is a finite group of automorphisms of a field $E$. Form the ring $E(G)$ which is the subring of the ring of endomorphisms of $E$ generated by $G$ and multiplication by elements of $E$. Note that if $\lambda\in E$ then the composite $g\cdot \mu_\lambda$ equals the composite $\mu_{g(\lambda)} \cdot g$. Show that the $E$-subalgebras of the group ring $E(G)$ are those of the form $E(H)$ for $H$ a subgroup of $G$.

Note By an $E$-subalgebra we mean those subrings that contain all the multiplications by elements $\lambda \in E$.

# Spooky geometry II

Find a compact metric space that does not embed in $\Bbb{R}^n$ for any $n$.

# Compact isometries

As seen in this post, any isometry from a compact metric space $K$ to itself is surjective, so the set $I(K)$ of all such isometries is actually a group.

Give $I(K)$ a reasonable metric and prove that it is compact as well. What can one say about the sequence of iterates $K, I(K), I(I(K)), \dots$?

# Duality in polynomial rings

Let $A$ be a polynomial ring over a commutative ring (!) $k$ in $n$ variables. Show that for any $A$-module $M$ and $0\leqslant q\leqslant n$, there are natural isomorphisms

$\mathrm{Tor}_q^A(k,M) \simeq \mathrm{Ext}_A^{n-q}(k,M)$

# Balanced signs

Assume $(p_n)$ is a nonincreasing sequence of positive integers whose sum $p_1+p_2+\cdots$ diverges and assume signs $(\varepsilon_n)$ are chosen so that the modified series $\varepsilon_1p_1+\varepsilon_2p_2+\cdots$ converges. Then

$\liminf\limits_{n\to\infty}\dfrac{\varepsilon_1+\cdots+\varepsilon_n}n \leqslant 0\leqslant \limsup\limits_{n\to\infty}\dfrac{\varepsilon_1+\cdots+\varepsilon_n}n$

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

$\lim\limits_{n\to\infty} (\varepsilon_1+\cdots+\varepsilon_n)p_n=0$