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\newcommand{\Z}{\mathbb{Z}}
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\begin{document}
\prof{George McNinch}
\quizdate{\today}
\class{Math 222}
\examname{Practice Problems 3}
\heading

\begin{problem}
  Give definitions for the following terms: prime, unit, group, field.
\end{problem}

\begin{problem}
  Let $p>2$ be a prime number, and let $a,b \in \Z$. Assume that $ab =
  2 + kp$ for some $k \in \Z$. Show that either
  $a \not \equiv 1 \pmod p$ or
  $b \not \equiv 1 \pmod p$ (or both, of course).
\end{problem}

\begin{problem}
  Factor the following into primes in $\Z[i]$:
  $\alpha = 221$, $\alpha = 3 + 5i$.
\end{problem}

\begin{problem}
  Factor the following into primes in $\Z[\sqrt{2}]$:
  $\alpha = 221$, $\alpha = 3 + 5 \sqrt{2}$.
\end{problem}

\begin{problem}
  Suppose that $\alpha \in \Z[\sqrt{2}]$ is prime. Prove that the
  conjugate $\overline{\alpha}$ is also prime.
\end{problem}

\begin{problem}
  Consider the ring $R = \Z[\sqrt{-5}]$.
  \begin{subproblem}
    Define the norm $N(\alpha)$ of an element $\alpha$ of this ring.
    What properties do you expect the norm to possess?  Prove that
    $N(\alpha) \in \Z$ for every $\alpha = a + b \sqrt{-5} \in R$
  \end{subproblem}
  \begin{subproblem}
    Compare the formula you gave for the norm of $\alpha = a +
    b\sqrt{-5}$ with the determinant of the following matrix:
    \begin{equation*}
    M_\alpha = \begin{bmatrix}
      a  & -5b \\
      b  & a \\
    \end{bmatrix}
  \end{equation*}
  Now prove that $N(\alpha \cdot \beta) = N(\alpha) \cdot N(\beta)$.
  Hint: Compare $M_{\alpha \cdot \beta}$ with $M_\alpha \cdot M_\beta$.
  \end{subproblem}
  \begin{subproblem}
    Prove that $\alpha \in R$ is a unit if and only if $N(\alpha) = \pm 1$.
    Can the norm ever be negative for this ring?
  \end{subproblem}
  \begin{subproblem}
    Compute the units in $R$.
  \end{subproblem}
  \begin{subproblem}
    Prove that if $N(\alpha) = p$ is a prime integer, then
    $\alpha$ is a prime element in $R$.
  \end{subproblem}
\end{problem}

\begin{problem}
  We proved in class that $\Z_p[i]$ is a field whenever $p \equiv
  3 \pmod 4$; here $\Z_p[i]$ consists of all elements $a + bi$ where
  $a$ and $b$ are regarded as coming from $\Z_p$.

  \begin{subproblem}
    Show that $\Z_{13}[i]$ is not a field by finding a pair of
    zero-divisors.
  \end{subproblem}
  \begin{subproblem}
    Compute the order of $1 + i$ in $\Z_{7}[i]$. Show that the group
    of units in $\Z_7[i]$ is cyclic of order 48. You should try to do
    as few calculations as possible.  Note that you \emph{know} before
    starting that the order of any element divides 48, since there are
    only 48 non-zero elements in $\Z_7[i]$. So, to show that an
    element $x$ has order 48, one only needs to check that $x^k \not =
    1$ for any divisor $k$ of 48. In fact, you can probably cut down
    the number of checks necessary! See how few you can get away
    with...
  \end{subproblem}
  \begin{subproblem}
    Show that the polynomial $f(x) = x^2 - 4x + 5$ possesses a root in
    $\Z_{7}[i]$ (in fact, two roots) but has no roots in $\Z_{7}$.
    Hint: for candidates for roots of $f(x)$, look for elements $\alpha
    \in \Z_{7}[i]$ with $\alpha + \overline{\alpha} = 4$ and 
    $\alpha \cdot \overline{\alpha} = 5$.
  \end{subproblem}
\end{problem}

\begin{problem}
  Let $D_9$ be the group of symmetries of a regular $9$-gon, and let
  $R_9$ be the subgroup consisting of all rotations of the $9$-gon.
  \begin{subproblem}
    Find all subgroups of $R_9$. How many rotations generate the
    cyclic group $R_9$?
  \end{subproblem}
  \begin{subproblem}
    Let the vertices of the $9$-gon be numbered (clockwise)
    $1,2,\dots,9$, and consider the ``flip'' given by
    $$f_1=(2 9)(3 8)(4 7)(5 6)$$
    in cycle notation.
    Let $r$ be the ``clockwise'' rotation
    $$r = (1 2 3 4 5 6 7 8 9).$$
    Show that the symmetry $r^2 f_1
    r^{-2}$ has order 2 and leaves the vertex $3$ unmoved. (in fact,
    this is the flip $f_3$). What is $f_3(2)$? i.e. what vertex does
    this flip interchange with the vertex $2$?
  \end{subproblem}
  \begin{subproblem}
    Show that $f_1 r f_1$ is a rotation.  Compute its cycle
    representation.  Does this rotation generated $R_9$?
  \end{subproblem}
  \begin{subproblem}
    Find a non-Abelian (non-commutative) subgroup of $S_9$ of order 6
    as follows.  Let $R_9$ be the cyclic group of order 9 generated by
    the clockwise rotation of the $9$-gon: $(123456789)$. Find a
    subgroup $R_3 < R_9$ of order 3.  Then let $f = (14)(25)(36)$.
    Show that $f g = g^{-1} f$ for $g \in R_3$ and use this to
    construct your subgroup.  [Remark: this subgroup is actually ``the
    same'' as the group $D_3$. Do you see why?] [Another remark: this
    subgroup is \emph{not} a group of symmetries of the 9 gon.  Why
    not?]
  \end{subproblem}
\end{problem}

\begin{problem}
  Consider the group $S_8$, the symmetric group on $\{1,2,\dots,8\}$.
  \begin{subproblem}
    Write the permutation $$\sigma = \left (
    \begin{matrix} 
      1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
      2 & 3 & 1 & 4 & 8 & 7 & 6 & 5
    \end{matrix} 
  \right )$$
  in cycle notation.
\end{subproblem}
\begin{subproblem}
  Write the permutation $\sigma$ from (a) as a product of
  transpositions.
\end{subproblem}
\begin{subproblem}
  If you were to write the permutation $\sigma$ from (a) as a product
  of the particular transpositions
  $$\{(12),(13),(14),(15),(16),(17),(18)\}$$
  tell how many would be
  required. (You don't have to give me a minimal number here, just
  describe a procedure for writing $\sigma$ as such a product, and
  tell how many your procedure would require.)
\end{subproblem}
\begin{subproblem}
  Compute the sign of $\sigma$.  How many transpositions occurred in
  the expression in (b)? in (c)? Is $\sigma$ an element of $A_8$, the
  alternating group?
\end{subproblem}
\end{problem}

\begin{problem}
  Show that the group of units of $\Z_{16}$ is not cyclic.
  (Recall that the group of units of $\Z_{16}$ has order $\phi(16)$ where
  $\phi$ is Euler's function.)
\end{problem}

\begin{problem}
  Regard $D_7$ as a subgroup of $S_7$.  Let $r = (1234567)$ be the
  clockwise rotation of the $7$-gon.  Show that $\text{sign}(r) = 1$
  so that $r \in A_7$. Show that the elements of $D_7$ which lie in
  the alternating group $A_7$ are precisely the rotations of the
  $7$-gon. (Remark: a similar statement is true for $D_n$ whenever $n$
  is \emph{odd.} However, it is not true when $n$ is even!!)
\end{problem}

\end{document}
