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\noindent
\parbox{2in}{\bf Math 4720 \& 5210} 
\hfill {\Large \bf Test \#2: Ring Theory} \hfill
\parbox{2in}{\bf \hfill Due: April $17^{\text{th}}$, 2019}

\vspace{0.1in}

\noindent {\large \bf Name:} \underline{\hspace{3in}} \hfill \hfill {\bf \sc Show your work!!}

\vspace{0.1in}

\noindent {\large \bf 1. (12 $\times$ 5 = 60 points):} 
If the statement is always true, write ``True'' and prove it. 
If the statement is never true, write ``False'' and prove that it can never be true.
If the statement is true in some cases and false in others, write ``Possible'' then give an example and a counter-example.
\begin{enumerate}[(a)]
\item Let $n \in \mathbb{Z}$ and $n>1$.\\ 
        {\bf TRUE / POSSIBLE / FALSE}: Both cancellation laws hold in $\mathbb{Z}_n$.

\item Let $f(x) \in \mathbb{Q}[x]$ be a reducible polynomial of degree 5 with no roots in $\mathbb{Q}$.\\
        {\bf TRUE / POSSIBLE / FALSE}: $f(x)$ has no irreducible factor of degree 3.
        
\item Let $R$ and $S$ be commutative rings with $1$. In addition, $\mathrm{char}(R)=3$ and $\mathrm{char}(S)=2$.\\
        {\bf TRUE / POSSIBLE / FALSE}: $R \times S$ has characteristic 6.
        
\item Let $R$ be a sub{\bf{}ring} of $\mathbb{C}$.\\
        {\bf TRUE / POSSIBLE / FALSE}: $f(x)=6x+2$ is irreducible in $R[x]$.
        
\item Let $R$ be a commutative ring with $1$ and $I \triangleleft R$ a prime ideal such that $\diagquotient{R}{I}$ is finite.\\ 
          {\bf TRUE / POSSIBLE / FALSE}: $I$ is maximal.
          
\item $n \in \mathbb{Z}$ and $n>1$.\\
       {\bf TRUE / POSSIBLE / FALSE}: $x^3+1$ is irreducible in $\mathbb{Z}_n[x]$.
       
\item $n \in \mathbb{Z}$ and $n>1$.\\
        {\bf TRUE / POSSIBLE / FALSE}: $\mathbb{Z}_n[x]$ is a UFD but not a Euclidean domain.
        
\item Let $\mathbb{F}$ be a subfield of $\mathbb{C}$, $f(x)=x^2-5 \in \mathbb{F}[x]$, and $I=(f(x)) \triangleleft \mathbb{F}[x]$.\\
        {\bf TRUE / POSSIBLE / FALSE}: $\diagquotient{\mathbb{F}[x]}{I}$ is a field.
        
\item Let $f(x)=a(x)b(x)c(x)$ be a factorization in $\mathbb{Q}[x]$ of $f(x)$ into irreducibles such that $a(x),b(x),c(x)$ are distinct factors (they are not associates). Moreover, let $I=(f(x))$.\\
        {\bf TRUE / POSSIBLE / FALSE}: $\diagquotient{\mathbb{Q}[x]}{I}$ is a direct product of fields.
        
\item Let $\mathbb{F}$ be a subfield of $\mathbb{R}$, and $f(x) \in \mathbb{F}[x]$.\\
        {\bf TRUE / POSSIBLE / FALSE}: $f(x)$ is irreducible in $\mathbb{R}[x]$ but not in $\mathbb{F}[x]$.
        
\item Let $R$ be a sub{\bf{}ring} (with $1$) of $\mathbb{C}$.\\
        {\bf TRUE / POSSIBLE / FALSE}: $4$ is irreducible.
        
\item Let $R$ be a ring with $1$ whose characteristic is $7$.\\
        {\bf TRUE / POSSIBLE / FALSE}: $R$ is finite.
        
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\end{enumerate}

\noindent {\large \bf 2. (35 points):} Isomorphic or not?\\ Decide whether the following pairs of rings are isomorphic or not. Justify your answer.
\begin{enumerate}[(a)]
\item $\displaystyle{\left(\mathbb{Z}[x,y]\right)^{2 \times 2}}$ \quad ($2 \times 2$ matrices with entries in $\mathbb{Z}[x,y]$) \qquad and  \qquad $\mathbb{Q}(x)$ \quad (rational functions)

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\item The fraction field of $\mathbb{Z}[\sqrt{2}]$ \qquad and  \qquad $\diagquotient{\mathbb{Q}[x]}{(x^2-2)}$

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\item $\mathbb{Z}[i]$ \qquad and $\mathbb{Q}[x,y,z]$

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\item $\mathbb{Z}_3$ \qquad and \quad $\diagquotient{\mathbb{Z}[x]}{(x,3)}$

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\item $\mathbb{Z}_6[x]$ \qquad and \qquad $\mathbb{Z}_4[x,y]$

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\item For this last part, consider the ring of Gaussian integers $\mathbb{Z}[i]$ and the ring of integers $\mathbb{Z}$. Obviously these rings are not isomorphic. However, they have many matching (ring) properties [For example, they're both commutative]. List as many matching properties as you can. Also, give {\it at least} one property which doesn't match (showing that they cannot be isomorphic).

\end{enumerate}

\newpage

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\noindent {\large \bf 3. (10 points):} Show $\mathbb{Q}[\sqrt{3}]$ is a sub{\bf{}field} of $\mathbb{R}$. (First show it is a subring, then show it is a field.)  

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\noindent {\large \bf 4. (15 points):} Give examples of {\bf non-zero proper} ideals in $\mathbb{Z}[x]$ which are (i) maximal, (ii) prime but not maximal, and (iii) neither prime nor maximal. Can we find similar examples if we switch to $\mathbb{Q}[x]$?

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\noindent {\large \bf 5. (30 points):} Isomorphic fun! $\stackrel{\cdot\;\cdot}{\smile}$

\begin{enumerate}[(a)]

\item Let $S = \left\{ \begin{bmatrix} a & -b \\ b & a \end{bmatrix} \;\Big|\; a,b \in \mathbb{R} \right\}$. Show that $S$ is a subring of $\mathbb{R}^{2 \times 2}$ ($2 \times 2$ real matrices) which is isomorphic to $\mathbb{C}$ (the field of complex numbers).

\vspace{0.05in}

\item Prove $\diagquotient{\mathbb{Q}[x]}{(x^2-1)} \cong \mathbb{Q} \times \mathbb{Q}$ without using Chinese remaindering. Specifically use the map defined by $\varphi(f(x))=(f(-1),f(1))$ and apply the first isomorphism theorem.

\end{enumerate}

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\noindent {\large \bf 6. (10 points):} Let $\mathbb{F}$ be a field. First, explain why the only ideals of $\mathbb{F}$ are $\{0\}$ and $\mathbb{F}$ itself. Then show that if $\varphi:\mathbb{F} \to R$ is a homomorphism, then either $\varphi(x)=0$ for all $x \in \mathbb{F}$ or $\varphi$ is injective (i.e. one-to-one).

\vspace{0.15in}

\noindent {\large \bf 7. (35 points):} Workin' mod $n$.
\begin{enumerate}[(a)]
\item Factor $x^8-1$ in $\mathbb{Z}_2[x]$.

\vspace{0.05in}

\item Give {\bf examples} of irreducible polynomials of degrees 1, 2, and 3 in $\mathbb{Z}_5[x]$ (don't try to list out {\it all} of the irreducibles).

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\item Construct a field of order 9, call it $\mathbb{F}_9$. Write down the addition and multiplication tables for $\mathbb{F}_9$.
\end{enumerate} 

\vspace{0.15in}

\noindent {\large \bf 8. (15 points):} Let $R$ be a commutative ring with $1$. Recall that $R$ is Artinian if it has the descending chain condition on ideals (DCC). This means that given ideals $I_k \triangleleft R$ such that $I_1 \supseteq I_2 \supseteq \cdots$ there exists some positive integer $N$ such that $I_N=I_{N+1}=I_{N+2}=\cdots$ (every descending chain of ideals eventually stabilizes). 

Show that if $R$ is Artinian, then $R$ is classical (every element of $R$ is either zero, a zero divisor, or a unit). 

{\it Hint:} Let $x \in R$. Suppose that $x$ is not zero and not a unit. Consider $R \supseteq (x) \supseteq (x^2) \supseteq \cdots$.

\vspace{0.05in}

{\it Pointless note:} Instead of Artinian (i.e., our ring has the DCC) we could just assume descending chain condition on principal ideals (DCCP) holds. 

\vspace{0.2in}

\begin{center} \Large \bf Problems for 5210 Students. \end{center}

\noindent {\large \bf 9. (15 points):} Let $R$ be a principal ideal domain and $\{0\} \not= I \triangleleft R$. Show that $R/I$ is both Noetherian and Artinian (i.e., $R/I$ has both the ascending and descending chain conditions on ideals).

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\noindent {\large \bf 10. (20 points):} Let $R$ be a commutative ring with $1 \not=0$. We say that $R$ is a {\it local ring} if $R$ has a unique maximal ideal (i.e. $R$ has exactly one maximal ideal). 

\begin{enumerate}[(a)]

\item Show that if $R$ is local with maximal ideal $M$, then $R-M = \{ x \in R \;|\; x \not\in M\} = R^\times$ -- that is -- the units of $R$ are exactly the elements which are outside of $M$. 

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\item Now let $M=R-R^\times$ (the collection of non-units). Show that if $M$ is an ideal, then $R$ is a local ring with maximal ideal $M$ -- that is -- if the non-units form an ideal, then that ideal is the unique maximal ideal of $R$ so that $R$ is a local ring.

\end{enumerate} 

\noindent {\large \bf 11. (15 points):} You constructed a field of order $9$ (i.e., $\mathbb{F}_9$) in 7(c). Consider $Y^3+Y+2 \in \mathbb{F}_9[Y]$.
\begin{enumerate}[(a)]
\item Is $Y^3+Y+2$ irreducible in $\mathbb{F}_9[Y]$? Why or why not?
\item Identify the ring $\diagquotient{\mathbb{F}_9[Y]}{(Y^3+Y+2)}$
\end{enumerate}

\end{document}

