Spring 2014 MATH 415B Homework assignments:

You will find the following material useful. It contains a lab manual for the computer exercises using GAP.

1st Homework due Friday, 17th of January: Summarize what you have learnt in MATH 415A.

2nd Homework due Friday, 24th of January: Review Chapter 4 from Lay's book for MATH 215 and do problems 37 Section 23, 37 Section 26, 34, 37 Section 27 in Fraleigh's book.

3rd Homework due Friday, 31th of January: Download the computer algebra system GAP from www.gap-system.org and install it on a computer of your choice. If you need help, please talk to George Todd, the GTA for the course. Do problems 2, 10, 26 in Section 29, and 9,26,27 in Section 30 of Fraleigh's book.

1. Let $V$ be an $n$-dimensional $F$-vector space and let $V^{\star}$ be its dual space. For a subspace $W$ of $V$ define the annihilator $\text{Ann}(W)$ of $W$ as $\text{Ann}(W):=\{\lambda \in V^{\star} | \lambda(W) = 0 \}$. Show that
   1. $\text{Ann}(W)$ is a subspace of $V^\star$.
   2. If the dimension of $W$ is $k$, what is the dimension of $\text{Ann}(W)$?
   3. If $W_1, W_2$ are subspaces of $V$ with $W_1 \le W_2$ then $\text{Ann}(W_1) \ge \text{Ann}(W_2)$.
2. Give a proof of the Steinitz exchange theorem stated in class.
3. Let $R$ be a ring with identity. Use Zorn's Lemma to show that there exists a maximal proper ideal in $R$.
4. Do problem 27, Section 30 in Fraleigh's book.
5. Do the GAP exercise 17.2 a)-d), page 63, in the GAP lab manual

1. Let $E$ be a field extension of the field $F$ and let $A$ and $B$ be subsets of $E$. Show that $F(A)(B) =F(B)(A) = F(A \cup B)$. Show furthermore that in case $A=\{a_1,\ldots,a_n\}$ with $a_1,\ldots,a_n$ algebraic over $F$, then $F(A)=F[a_1,\ldots,a_n]$.
2. Do the GAP exercise 21.1 and 21.2, page 68, in the GAP lab manual
3. Do problems 26, 30, 31 in Section 31 in Fraleigh's book.

1. Do problems 27, 28, 29, 32, 36 in Section 31 in Fraleigh's book.
2. Read Section 33 in Fraleigh's book.

1. Let $E$ be a field extension of the field $F$ and let $A$ be a subset of $E$. Show that $F(A)$ is the union of all $F(B)$, where $B$ ranges over all finite subsets of $A$.
2. Let $F$ be a field and let $F(x)$ be the rational function field. Show that $y$ in $F(x)\setminus F$ is transcendental over $F$. Determine when $F(y) = F(x)$.
3. Do problems 8, 28, 32, 36, 38, 39 in Section 48 in Fraleigh's book.

8th Homework due Friday, 7th of March: Do problems 4,5,6, 11, 13 in Section 49 and problems 3,5 in Section 50 in Fraleigh's book.

10th Homework due Friday, 11th of April:

Let $n \in \mathbb{N}$, $T=\{1,\dots,n\}$, and let $S_n$ be the symmetric group on $T$. Show that the set $T \times T$ is an $S_n$-set if we define $\pi * (i,j) := (\pi(i),\pi(j))$ for $\pi \in S_n$ and $i,j \in T$. Determine the $S_n$-orbits of $T \times T$.

Do problems 11,12,13,16 in Section 53, problem 20 in Section 16, and problem 6 in Section 17 in Fraleigh's book.

11th Homework due Friday, 18th of April: Do problems 17, 18(e,f,g,h), 19, 22 in Section 53 and problem 8 in Section 54.

Do the GAP exercise 33.4 and 33.5, page 96, in the GAP lab manual Also choose 5 monic rational polynomials and compute the order of their Galois groups using the GAP-function GaloisType.

12th Homework due Friday, 25th of April: Do problems 5, 12 in Section 54, problems 13,15,19,22 in Section 36, and problems 4,5 in Section 37.