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Math 504 - Homework

• LaTeX-ed solutions are encouraged and appreciated.
• If you use LaTeX, hand-in a printed version of your homework.
• You are encouraged to discuss homework problems with classmates, but such discussions should NOT include the exchange or written material.
• Writing of homework problems should be done on an individual basis.
• Outside references for material used in the solution of homework problems should be fully disclosed.
• References to results from the textbook and/or class notes should also be included.
• The following lists should be considered partial and tentative lists until the word complete appears next to it.
• Use 8.5in x 11in paper with smooth borders. Write your name on top of each page. Staple all pages.


Problem Set 10 Due 05/08/2017 (complete)

1. Complete the proof of the following proposition. The first part was done in class. If $E/K$ is a Galois extension, and $F/K$ is any field extension, then $EF/F$ is a Galois extension. Moreover, $\gal(EF/F)$ embeds in $\gal(E/K)$, and when $E/E\intersection F$ is a finite extension, $\gal(EF/F) \isom \gal(E/E\intersection F).$
2. Show 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.
3. Show, by counterexample, that the finite hypothesis in the previous problem is necessary.
4. Let $l$ be a constructible straight line, $A$ a constructible point on $l$, and $\theta$ a constructible angle. Show that the straight line that goes through $A$ and forms an angle $\theta$ with $l$ is constructible.

Problem Set 9 Due 05/01/2017 (complete)

1. Prove that a directed union of algebraically independent sets over $K$ is algebraically independent over $K$. In particular, the union of a chain of algebraically independent sets over $K$ is algebraically independent over $K$.
2. Given $S\subseteq T$ with $S$ algebraically independent over $K$ and $F$ algebraic over $K(T)$, there is a transcendence basis $B$ with $S\subseteq B\subseteq T$. In particular, any field extension $F/K$ has a transcendence basis.
3. Prove the following version of the exchange property: Let $F/K$ be a field extension, $S,T\subseteq F$ be each algebraically independent over $K$, with $|S| < |T|$. There is $\beta\in T-S$ such that $S\union \{\beta\}$ is algebraically independent over $K$.
4. Prove that for a tower $L/F/K$, $tr.d._K(L) = tr.d._F(L) + tr.d._K(F)$
5. Prove that if $f(t_1,\dots,t_n)$ is a symmetric polynomial in variables $t_1,\dots,t_n$, there exists a polynomial $g$ such that $f(t_1,\dots,t_n)=g(s_1,\dots,s_n)$.

Problem Set 8 Due 04/21/2017 (complete)

1. Let $G$ be a group, and $H_1,H_2\leq G$. Show that $[G:H_1\intersection H_2] \leq [G:H_1][G:H_2].$
2. Prove that the normal closure of a finite separable extension over $K$ is a finite Galois extension.
3. Given a projective system $(G_i|i\in I)$ of groups, with maps $(\rho_{i,j}|i\leq j)$, show that the following subset of the product $\left\{a\in\prod_{i\in I}G_i\middle|\rho_{i,j}(a_j)=a_i \text{ for all } i\leq j\right\},$ together with the projections on the factors is a projective limit for the system.