### Sidebar

people:fer:504ws:504ws_old_homework

## Math 504 - Old 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 3 Due 02/09/2018 (complete)

1. Prove the corollary stated in class: If $K\leq E_i\leq F$ and each $E_i/K$ is algebraic, then the join $\displaystyle\bigjoin_{i\in I}E_i$ is algebraic over $K$.
2. Let $F/K$ be a finite extension. Prove that every endomorphism of $F$ that fixes $K$ is an automorphism of $F$.
3. Consider the extension $F=\Q(\alpha,\omega)$ of $\Q$ discussed in class, where $\alpha$ is a root of $x^3-2$ and $\omega$ is a root of $x^2+x+1$. Construct several automorphisms of $F$. Is there a bound for the number of automorphisms of $F$?

Problem Set 2 Due 02/02/2018 (complete)

1. Show that the direct (cartesian) product of two fields is never a field.
2. Show that $\Q(\sq{2})\neq\Q(\sq{3})$. Generalize.
3. Page 163, IV.2.1
4. Page 163, IV.2.2, IV.2.4

Problem Set 1 Due 01/26/2018 (complete)

1. Let $G$ be a group and $N\normaleq G$. $G$ is solvable iff $N$ and $G/N$ are solvable. In this case, $$l(G) \leq l(N) + l(G/N)$$
2. If $L$ is a poset in which every subset has a l.u.b., then every subset of $L$ also has a g.l.b.
3. Given a lattice $(L,\meet,\join)$ in the algebraic sense, show that the binary relation defined by $$x \leq y \quad iff \quad x \meet y = x$$ is a partial order on $L$, and for $x,y \in L$, $x \meet y$ is the g.l.b.{x,y}, and $x \join y$ is the l.u.b.{x,y}.
4. Let $A$ be a universal algebra, and $\sub(A)$ the complete lattice of subuniverses of $A$. If $D\subseteq\sub(A)$ is directed, then $\ds\left(\bigunion_{X\in D}X\right)\in\sub(A)$.