Algebra General Exam 2004

August 16, 2004, UVA

Problem 1

( 10 points) Assume that the group $G$ is a direct product of two finite subgroups $A$ and $B, G=A \times B$, and that $|A|,|B|$ are relatively prime. Show that $H= (H \cap A) \times(H \cap B)$ for any subgroup $H$ of $G$.

Problem 2

( 14 points) Let $G$ be a group of order 56, and let $P_{p}$ for $p \in\{2,7\}$ be a Sylow $p$ subgroup of $G$.
(a) (6 points) Show that $P_{2}$ or $P_{7}$ is normal in $G$.
(b) ( 3 points) Give an example of a group $G$ with $|G|=56$ where $P_{2}$ is not normal.
(c) ( 5 points) Show that there exists a group $G$ of order 56 with a non-normal $P_{7}$. You can either do this by exhibiting a concrete example with this property or by describing how to construct such a group. In the latter case you have to justify why your approach works but you needn't give all details of the construction.

Problem 3

( 12 points, 6 points each) Consider the ring $R=\mathbb{Z}[\sqrt{-7}]=\{m+n \sqrt{-7} \mid m, n \in \mathbb{Z}\}$.
(a) Is $R$ a UFD? Give arguments for your answer.
(b) Exhibit an ideal $I$ in $R$ which is not principal. Show that your $I$ is not principal.

Problem 4

(12) Let $L / K$ be a finite Galois extension. Suppose there exists an element $\alpha \in L$ and another root $\alpha^{\prime}$ of the minimal polynomial $\mu_{\alpha \mid K}$ of $\alpha$ over $K$ such that the difference $\alpha^{\prime}-\alpha$ is an element of $K \backslash\{0\}$.
(a) (9 points) Prove that the characteristic $p$ of $K$ is different from 0 and that $p$ divides $[L: K]$.
(b) ( 3 points) Give an example of an extension $L / K$ and elements $\alpha, \alpha^{\prime}$ as described above.

Problem 5

( 10 points, 5 points each) Let $N$ be the $\mathbb{Z}$-submodule of $\mathbb{Z}^{3}$ generated by the column vectors $(2,2,-2)^{t},(-4,-2,4)^{t}$ and $(2,4,4)^{t} \in \mathbb{Z}^{3}$.
(a) Determine the structure of the abelian group $\mathbb{Z}^{3} / N$.
(b) Determine a basis $y_{1}, y_{2}, y_{3}$ of $\mathbb{Z}^{3}$ and natural numbers $d_{1}\left|d_{2}\right| d_{3}$ such that $d_{1} y_{1}, d_{2} y_{2}, d_{3} y_{3}$ is a $\mathbb{Z}$-basis of $N$.

Problem 6

( 10 points) Let $K$ be an arbitrary field ( $\rightarrow$ case distinction!). Classify, up to similarity, all matrices $A \in G L_{4}(K)$ of order 2. (Use an appropriate canonical form.)

Problem 7

( 12 points) Consider the group $G=S L_{2}\left(\mathbb{F}_{4}\right)$.
(a) ( 4 points) Show without specifying any matrix that $G$ contains an element of order 5 .
(b) ( 8 points) Exhibit a concrete matrix $A \in S L_{2}\left(\mathbb{F}_{4}\right)$ of order 5 . Use $\mathbb{F}_{4}= \left\{0,1, \alpha, \alpha^{2}\right\}$, where $\alpha$ is a root of $x^{2}+x+1$. Describe in detail how you obtained $A$; you shouldn’t just guess!
(Hint: You might first factorize $x^{5}-1$ in $\mathbb{F}_{4}[x]$.)

Problem 8

(8 points) Let $K$ be a field, $V$ a finite-dimensional vector space over $K$, and $v, w$ two non-zero vectors in $V$. Show that $v \otimes w=w \otimes v$ in $V \otimes_{K} V$ if and only if there exists a $c \in K^{*}$ such that $w=c v$.

Problem 9

( 12 points, 3 points each) Decide in each of the following four cases whether the given statement is true or false. You need not give any arguments.
(a) If $n \geq 3$ is a natural number and $A=J_{n}(0) \in M_{n}(\mathbb{R})$ is a Jordan block matrix of size $n \times n$ with eigenvalue 0 , then $J_{2}(0) \oplus J_{n-2}(0)$ (i.e. two Jordan blocks, one of size 2 and one of size $n-2$ ) is the Jordan canonical form of $A^{2}$.
(b) If $R$ is an integral domain which is also a finite-dimensional $K$-algebra for some field $K \subseteq R$, then $R$ is itself a field.
(c) Let $L / K$ be a field extension, and assume that $L$ contains a primitive $n^{\text {th }}$ root of unity $\zeta_{n}$. Then $M / K$ is normal for any subfield $M$ of $K\left(\zeta_{n}\right)$ which contains $K$ (i.e. we are considering the tower $\left.L / K\left(\zeta_{n}\right) / M / K\right)$.
(d) Let $V$ be a finite-dimensional vector space over a field $K$ and $B: V \times V \rightarrow K$ a symmetric $K$-bilinear form. If there exists a subspace $W \neq\{0\}$ of $V$ such that $V=W \oplus W^{\perp}$, where $W^{\perp}:=\{x \in V \mid B(x, w)=0$ for all $w \in W\}$, then $B$ is nondegenerate.