Problem Set 13 Due 05/07/2018 (complete)

1. Let $K$ and $L$ be fields. Show that the set $\hom(K,L)$ of all homomorphisms from $K$ to $L$, is linearly independent over $L$. In particular $\aut(K)$ is linearly independent over $K$.
2. Prove that a finite group $G$ is solvable iff there is a finite sequence of subgroups $$1=H_0\leq H_1 \leq \cdots \leq H_{n-1} \leq H_n=G$$ such that each $H_i\normaleq H_{i+1}$ and $H_{i+1}/H_i$ is cyclic. Show, with a counterexample, that this equivalence does not hold in general for arbitrary groups.
3. Define: an angle $\theta$ is constructible if there are two constructible straight lines forming an angle $\theta$.
   Prove: let $l$ be a constructible straight line, $A$ a constructible point on $l$, and $\theta$ a constructible angle. The straight line(s) that go through $A$ and form an angle $\theta$ with $l$ is(are) constructible.

Problem Set 12 Due 04/27/2018 (complete)

1. Let $F/K$ be a field extension, $S\subseteq T\subseteq F$ with $S$ algebraically independent over $K$, and $F$ algebraic over $K(T)$. Prove that there is a transcendence basis $B$, for $F$ over $K$, such that $S\subseteq B\subseteq T$. (Hint: prove that a directed union of algebraically independent sets over $K$ is algebraically independent over $K$, and use Zorn's lemma)
2. Let $F/K$ be a field extension and $S\subseteq F$. Prove that TFAE:
   1. $S$ is maximal algebraically independent over $K$,
   2. $S$ is algebraically independent over $K$ and $F$ is algebraic over $K(S)$,
   3. $S$ is minimal such that $F$ is algebraic over $K(S)$.
3. Let $F/E/K$ be a field tower. Prove that $$tr.d._K(F)=tr.d._E(F)+tr.d._K(E)$$
4. Let $K$ be a field, and $t_1,\dots,t_n$ independent variables. If $f(t_1,\dots,t_n)\in K[t_1,\dots,t_n]$ is a symmetric polynomial in variables $t_1,\dots,t_n$, there is a polynomial $g$, such that $$f(t_1,\dots,t_n) = g(s_1,\dots,s_n).$$ (Hint: Use double induction on $n$ and $d$, the total degree of $f$)

Old Homework