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\pagestyle{myheadings} \markright{Math 3340 (Spring 2021) \hspace{2in}
(HW8)}

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  {\large Homework \# 8 $\quad$ Math 3340 Spring 2021

    \medskip
    
    Due at the end of the day Friday, April 16, at midnight.
  }

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Please submit your completed homework via gradescope on canvas.
I encourage you to work with your classmates on this homework (except for the journal entries!)
When you submit your work, please {\bf list your collaborators}.
(Your grade will not be affected.)
Even if you work in a group, you should write up 
your solutions {\bf yourself}! 
You should include all computational details, and proofs should be carefully 
written with full details.
As always, please write {\bf neatly and legibly} (feel free to use \LaTeX\ to write up your solutions, if you wish!).

\vskip 4mm

%% END PREAMBLE

%%   \noindent {\bf Reading.} {\em The Secret to Raising Smart Kids} by Carol Dweck, in Scientific American.  You can find it
%% by Google, or in the files section on our canvas site (it is also linked to from the syllabus).

%% \vskip 0.1in

\noindent {\bf Journal entry.} There is no journal entry this week.

\vskip 0.1in

\noindent {\bf Exercises.} 

\vskip 0.1in

\begin{enumerate}

\item Let $R$ and $S$ be commutative rings.  Define addition and multiplication operations on $R \times S$,
  that make $R \times S$ into a ring.  What are the 0 and 1 elements in this ring?

\item Suppose that $R$ and $S$ are commutative rings, and that $f : R \to S$ is a function.
  A homomorphism of rings is a function that preserves addition and multiplication, and also sends 1 to 1:
  i.e. $f$ is a homomorphism of rings if for all $a, b \in R$,
  (a)\ $f(1) = 1$, (b)\ $f(a+b) = f(a) + f(b)$, and (c)\ $f(ab) = f(a) f(b)$.
  An isomorphism is a bijective homomorphism.  
  \babc
\item If $R$ and $S$ are fields, show that to check that $f : R \to S$ is a homomorphism of rings,
  we do not need to check $f(1) = 1$.
\item Show that if $f : R \to S$ is a homomorphism of rings, and $R, S$ are fields,
  then $f$ is always injective.
\item If $f : R \to S$ is a homomorphism of rings, show that $im(f)$ is a subring of $S$.
  Is it true that $f$ is injective if and only if $\ker(f) := \{ a \in R \mid f(a) = 0 \}$ is equal to
  $\{ 0 \}$?  If so prove it, if not, give a counter example.
\item Suppose that $m$ and $n$ are relatively prime.
  Define a function $f : \Z_{mn} \to \Z_n \times \Z_m$ by $f([a]) = ([a]_m, [a]_n)$.
  Show that $f$ is well-defined and an isomorphism of rings.
\item Show that if $a, b$ are two integers, then there exists an integer $c$ such that $c \equiv a \bmod m$ and that
  $c \equiv b \bmod n$.
  \eabc
  
\item Recall that the Euler $\phi$ function is $\phi(n)$ is the number of elements $1 \le m \le n$
  which are relatively prime to $n$.
  \babc
\item If $p$ is a prime number and $\alpha \ge 1$ is a positive integer, compute $\phi(p)$, and more generally, $\phi(p^\alpha)$.
\item If $m$ and $n$ are relatively prime, show that $\phi(mn) = \phi(m) \phi(n)$. (Recall from problem 2(d) that $\Z_{mn} \cong \Z_m \times \Z_n$ if
  $m$ and $n$ are relatively prime).
  \item Find a formula for $\phi(n)$, where $n = p_1^{\alpha_1} p_2^{\alpha_2} \ldots p_r^{\alpha_r}$.
  \eabc

\item For the following sets, with the standard addition and multiplication, determine whether they are
  (a) a field, (b) a commutative ring, (c) a noncommutative ring, (d) an Abelian group under multiplication, (e) A non-Abelian
  group under multiplication. (Each one can be several of these, perhaps).
  \babc
%% \item $\Q(\sqrt{5}) = \{ a + b \sqrt{5} \mid a, b \in \Q \} \subset \R$.
\item $\Z[\sqrt{5}] := \{ a + b \sqrt{5} \mid a, b \in \Z \} \subset \R$.
%% \item The set of all $2 \times 2$ matrices whose entries are in a field $F$ (you may assume that matrix multiplication and addition
%%   are both associative, and the distributive property holds).
\item If $F$ is a field, the set
  \[ H = \left\{ \begin{bmatrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{bmatrix} \mid a, b, c \in F \right\} \subseteq F^{3 \times 3}\]
  %% \item The set $R^x = \{ a \in R \mid \mbox{there exists $x \in R$ such that $ax = 1$}\}$, where $R$ is a commutative ring.
  \eabc

\item
  \babc
\item Find all monic irreducible polynomials of degree at most 3 in the polynomial ring $\Z_2[x]$.
  \item Show that the product of all irreducible polynomials of degree $\le 2$ is $x^4-x$ (or, what is the same: $x^4 + x$).
\item Find the complete factorization in $\Z_2[x]$ of $x^5 + x^4 + 1$.
  \eabc
  
\item
  \babc
\item Find all monic irreducible polynomials of degree at most 3 in the polynomial ring $\Z_3[x]$ (it might be easier to use the elements
  of $\Z_3$ as -1, 0, 1, instead of 0, 1, 2).
\item Find the complete factorization in $\Z_3[x]$ of $x^4 + 2x^2 + 2x + 2$.
  \eabc

\item Consider the polynomial $f = x^4+1$, over various fields $F$.
  \babc
\item Is $x^4+1$ irreducible in $\R[x]$?  If so, prove it, if not, find a factorization.
  \item Factor completely $x^4+1$ in $\C[x]$.
\item Suppose that $p \equiv 1 (\bmod 4)$.  Show that $x^4+1$ is a product of
  2 monic factors and so is not irreducible.
\item (Challenge, not part of HW):
  Show that $x^4+1$ has a proper factorization in $\Z_p[x]$, for every prime number $p$.
  \eabc

\item Let $F$ be a field, and consider the polynomial ring $F[x]$.
  In this problem, you may use as proved what we have covered in lecture 27
  (i.e. division algorithm, ideals, gcd, writing $d = gcd(f,g)$ as $d = af + bg$).
  \babc
\item Show that if $f$ divides $gh$, and $gcd(f, g) = 1$, then $f$ divides $h$.
\item
    Show that if $f \in F[x]$ is a non-zero polynomial, then there exists
    an element $a \in F$, and monic irreducible polynomials $p_1, p_2, \ldots, p_r$ in $F[x]$, and positive
    integers $\alpha_1, \alpha_2, \ldots, \alpha_r$ (for some integer $r \ge 0$), such that
    \[ f = a p_1^{\alpha_1} p_2^{\alpha_2} \ldots p_r^{\alpha_r}. \]
    \item 
      Show that any such factorization is unique, up to reordering the factors.
      (Hint: use induction on the degree of the polynomial.  If an irreducible polynomial appears
      in one factorization, does it also appear in the other factorization?)
    \eabc
\end{enumerate}
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