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\begin{document}
\title{MATH 172, Fall 2017, Take-home midterm}

\maketitle


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\begin{center}
{\bf Please print your name clearly!}
\end{center}
\bigskip

\noindent
{\bf Name:\underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }}
\bigskip

\noindent
{\it This test is due on Wednesday, 10/25/17, in class. While completing it, feel free to use your lecture notes from class, as well as your textbook. You are however not allowed to consult with anyone: it is understood that solutions to this midterm represent solely your own work with no outside assistance. Good luck!}
\bigskip

\noindent
{\bf Problem 1. (30 points)} Let $R$ be an integral domain and $F$ its fraction field. $R$ is called a {\it valuation domain} if for every nonzero element $\alpha \in F$, either $\alpha \in R$ or $\alpha^{-1} \in R$. Throughout this problem, assume that $R$ is a valuation domain.
\smallskip

{\bf Part a (10 points).} Let $I$ and $J$ be ideals in $R$. Prove that either $I \subseteq J$ or $J \subseteq I$. Conclude that $R$ has a unique maximal ideal (recall that rings with this property are called local rings).
\smallskip

{\bf Part b (10 points).} Suppose an ideal $I$ in $R$ is finitely generated. Prove that $I$ must be principal.
\smallskip

{\bf Part c (10 points).} Let $0 \neq \alpha \in F$ be such that
$$\alpha^n + a_{n-1} \alpha^{n-1} + \dots + a_1 \alpha + a_0 \in R$$
for some $a_{n-1},\dots,a_1,a_0 \in R$, $n \geq 1$. Prove that $\alpha \in R$.

\newpage



\noindent
{\bf Problem 2. (40 points)} Let $R = \zed[x]$, the ring of polynomials in one variable $x$ with integer coefficients.
\smallskip

{\bf Part a (20 points).} Prove that maximal ideals in $R$ are of the form $M = \left< p, f(x) \right>$, where $p \in \zed$ is a prime number and $f(x) \in R$ is a polynomial, which is irreducible in $\left( \zed/p\zed \right)[x]$.
\smallskip

{\bf Part b (10 points).} Prove that the ideal $I = \left< 3, x^2+x+1 \right>$ in $R$ is not prime. Since maximal ideals in $R$ are prime, $I$ must not satisfy the description in part a. Explain how.
\smallskip

{\bf Part c (10 points).} Is every nonzero prime ideal in $R$ maximal? If yes, prove your answer; if no, give a counterexample.



\newpage


\noindent
{\bf Problem 3. (30 points)} Let $R$ be an integral domain and let $I_1,I_2,\dots$ be a sequence of ideals in $R$.
\smallskip

{\bf Part a (10 points).} Suppose that for some $N$,
$$I_1 \cap I_2 \cap \dots \cap I_N = \{0\}.$$
Prove that at least one of $I_n$ for $1 \leq n \leq N$ must be $\{0\}$.
\smallskip

{\bf Part b (10 points).} Give an example of $R$ and a sequence of ideals such that
$$\bigcap_{n=1}^{\infty} I_n = \{0\}$$
while none of the ideals $I_n$ is equal to $\{0\}$.
\smallskip

{\bf Part c (10 points).} Assume that
$$\bigcap_{n=1}^{\infty} I_n = \{0\}$$
in $R$ and $I_n \neq \{0\}$ for any $n \geq 1$. Then prove that for every $m \geq 1$,
$$\bigcap_{n=m}^{\infty} I_n = \{0\}.$$


\newpage


\noindent
{\bf Problem 4. (40 points)} Let $R$ be a commutative ring with identity. $R$ is called {\it Noetherian} if every ascending chain of ideals
$$I_1 \subseteq I_2 \subseteq I_3 \subseteq \dots$$
stabilizes, i.e. there exists some $n \geq 1$ such that $I_n = I_{n+1} = I_{n+2} = \dots$ $R$ is called {\it Artinian} if every descending chain of ideals
$$\dots \subseteq I_3 \subseteq I_2 \subseteq I_1$$
stabilizes. 
\smallskip

{\bf Part a (10 points).} Is $\zed$ a Noetherian ring? Is it Artinian? Prove your answers without relying on the later parts of this problem.
\smallskip

{\bf Part b (10 points).} Prove that an Artinian integral domain must be a field.
\smallskip

{\bf Part c (10 points).} Define the {\it length} of a chain of $n+1$ prime ideals
$$P_0 \subsetneq P_1 \subsetneq P_2 \subsetneq \dots \subsetneq P_n$$
in a commutative ring $R$ to be $n$ (i.e., length is the number of inclusions instead of the number of ideals). {\it Krull dimension} of $R$, denoted $\dim(R)$, is the supremum of lengths of all chains of prime ideals in $R$. Prove that a commutative Artinian ring $R$ must have $\dim(R)=0$.

{\bf Part d (10 points).} Suppose $R$ is a PID, but not a field. Prove that $R$ is Noetherian and $\dim(R)=1$. Conclude that it cannot be Artinian.




\newpage

\noindent
{\bf Problem 5 (15 points).} Let $D \in \zed$ be squarefree, and recall the definition of the corresponding quadratic integer ring:
$$\zed[\sqrt{D}] = \left\{ a+b\sqrt{D} : a,b \in \zed \right\},$$
and let the norm on it be given by the field norm on $\que(\sqrt{D})$, as usual:
$$N(a+b\sqrt{D}) = \left| (a+b\sqrt{D}) (a-b\sqrt{D}) \right| = \left| a^2-Db^2 \right|,$$
for each $a+b\sqrt{D} \in \zed[\sqrt{D}]$. Recall also that this norm is multiplicative:
$$N(xy) = N(x)N(y),\ \forall\ x,y \in \zed[\sqrt{D}].$$
\smallskip

{\bf Part a (10 points).} Prove that an element $x \in \zed[\sqrt{D}]$ is a unit if and only if $N(x)=1$. Use this fact to prove that the only units in $\zed[i]$ are $\pm 1, \pm i$.
\smallskip

{\bf Part b (5 points).} Suppose $n > 1$ is an integer such that $n=a^2+b^2$ for some $a,b \in \zed$. Prove that $n$ is not prime in $\zed[i]$.
\bigskip

\noindent
{\bf Remark:} Notice that there can be a prime $p \in \zed$ such that $p = a^2+b^2$ for some $a,b \in \zed$ (for instance, $5 = 2^2 + 1^2$), but according to Part b it is no longer prime in $\zed[i]$. This effect is called {\it splitting} of a prime in an extension ring.


\newpage

\noindent
{\bf Problem 6 (10 points).} 
\bigskip

{\bf Part a (5 points).} Let $F$ be a field, and let $p(x) \in F[x]$ be an irreducible polynomial of degree 2. We know that the extension field $K$ of $F$ defined by $K = F[x] / \left< p(x) \right>$ contains at least one root of $p(x)$. Prove that in fact it contains all roots of $p(x)$.
\smallskip

{\bf Part b (5 points).} Prove that the polynomial
$$p(x) = 12x^8+15x^7+21x^5+6x^4-18x^3+2x^2+4x+1$$
is irreducible in $\que[x]$.



\newpage

\noindent
{\bf Problem 7 (10 points).} Let $R$ be an integral domain, and define $R[x,y]$ be the ring of polynomials in two variables $x$ and $y$ with coefficients in $R$. For which choices of $R$ is $R[x,y]$ a Euclidean domain? Prove your answer.




\end{document}