Posted by: holdenlee | May 25, 2010

## B7s: May 15, 2010

Problems

1. “Last night’s temperature took all values between -3 and 5.” Show that it suffices to say “Both -3 and 5 occured among last night’s minimum temperatures (by location).” Assume that temperature is a continuous function of time and location.
2. Let $H\subseteq \mathbb{R}, H\neq \{0\}$ be closed under addition. Let $f:H\to\mathbb{R}$ be nonincreasing and additive ($f(x+y)=f(x)+f(y);x,y\in H$). Show that $f(x)=cx$ on $H$ where $c\geq 0$.
3. Let $A,B$ be (additive) abelian groups. For $\nu, \chi$ homomorphisms from $A$ to $B$, define their sum by $(\nu+\chi)(a)=\nu(a)+\chi(a)$. Then the set of homomorphisms from $A$ to $B$ forms an abelian group. Suppose $A$ is a $p$-group ($p$ prime).
1. Prove that $H$ becomes a topological group under the topology by taking $p^kH,k\in \mathbb{N}$ to be a neighborhood base around 0.
2. Prove that $H$ is complete in this topology.
3. Prove that every connected component consists of a single element.
4. When is $H$ compact?
4. For what $x\in \mathbb{C}$ does there exist a proper subfield $\mathbb{F}\subset \mathbb{C}$ such that $\mathbb{F}(x)=\mathbb{C}$?

Solutions (Sketches)

1. Use intermediate value theorem twice, once for time and once for location.
2. Proof follows the same argument as in $\mathbb{R}$ (Cauchy’s Equation).
1. Show that negation is a continuous function $H\to H$ and addition is a continuous function $H\times H\to H$. The first is obvious. For the second, note that if $\alpha+\beta=\gamma$ then $(\alpha +p^kH)+(\beta+p^kH)\subseteq \gamma+p^kH$$((\alpha +p^kH),(\beta+p^kH))$ is an open set in $H\times H$.
2. Take the metric defined by $d(\nu,\chi)=p^{-l}$ where $l= -\max \{k \mid \nu - \chi \in p^kH\}$. Given a Cauchy sequence $\{\chi_i\}$, we define $\chi$ as follows: for $a$ with order $p^k$ by the Cauchy condition $\chi_i$ will eventually stay in the same coset $\alpha+p^kH$. Then $\chi_i(a)$ stabilizes; set $\chi(a)$ equal to this value. One can show that $\chi$ is a homomorphism and $\chi_i\to \chi$.
3. It suffices to show that each open set is disconnected; it suffices to show that $p^kH$ can be written as a union of disjoint closed sets. Just write $p^kH=p^{k+1}H\cup (p^kH-p^{k+1}H)$. (the complements of either can be written as a union of cosets=open sets in the form $\alpha+p^iH$.)
4. If $[p^kH:p^{k+1}H]$ is finite for each $k$, then the proof of compactness is similar to the proof in $\mathbb{R}^n$. Suppose there’s an open cover without a finite subcover; by induction and Pigeonhole Principle take a coset $\alpha_k+p^kH$ without a finite subcover. Take $\chi_k\in \alpha_k+p^kH$; this Cauchy sequence converges to some $\chi$ by completeness. There must be an open set in the cover containing $\chi$, but it contains $\alpha_k+p^kH$ for large enough $k$, contradiction.
Else $[H:p^kH]=\infty$ for some $k$. Then the cosets of $p^kH$ form an open cover without a finite subcover.
This reduces the problem to a group theory problem…
3. Working on it.