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\centerline{\titlefont Practice Problems for Second Midterm Exam}
\smallskip
\centerline{Modern Algebra, Dave Bayer, March 29, 1999}
\medskip

\bigskip

\centerline{This first set of problems are collected from various materials
already posted on the web.}

\bigskip
 
\prob
Working in the Gaussian integers $\ZZ[i]$, factor $2$ into primes.

\prob
Let $a=3-i$ and $b=2i$ be elements of the Gaussian integers $\ZZ[i]$.
\item{\bf (a)}  Find $q_1$, $c\in\ZZ[i]$ so $a=q_1\,b+c$ with
$\abs{c}<\abs{b}$. \item{\bf (b)} Now find $q_2$, $d\in\ZZ[i]$ so $b=q_2\,c+d$
with $\abs{d}<\abs{c}$.
\item{\bf (c)} Express $(a,b)\subset\ZZ[i]$ as a principal ideal.

\prob
The ideal $I=(2,1+3i)\subset\ZZ[i]$ is principal, where $\ZZ[i]$ are the
Gaussian integers. Find a single generator for $I$. (Repeat for
$I=(3,1+i)$, and $I=(6,3+5i)$.)

\prob
\item{\bf(a)} Prove that every positive integer can be uniquely factored
into primes, up to the order of the primes.
\item{\bf(b)} How do you need to modify this proof so it works for a
polynomial ring in one variable over a field?

\prob
Let $R$ be a principal ideal domain, and let
$$I_1 \quad\subset\quad I_2 \quad\subset\quad I_3
 \quad\subset\quad \cdots \quad\subset\quad I_n \quad\subset\quad \cdots$$
be an infinite ascending chain of ideals in $R$. Show that this chain {\it
stabilizes\/}, i.e.
$$I_N \quad=\quad I_{N+1} \quad=\quad I_{N+2} \quad=\quad \cdots$$
for some $N$.

\prob
Prove the {\it Eisenstein criterion\/} for irreducibility: Let $f(x)=a_n
x^n + \ldots + a_1 x + a_0 \in \ZZ[x]$, and let $p$ be a prime. If $p$
doesn't divide $a_n$, $p$ does divide $a_{n-1}, \ldots, a_0$, but $p^2$
doesn't divide $a_0$, then $f(x)$ is irreducible as a polynomial in
$\QQ[x]$.
\item{\bf(a)} First, what does $f(x)$ look like mod $p$?
\item{\bf(b)} Now, suppose that there is a nontrivial factorization
$f(x)=g(x)h(x)$ in $\ZZ[x]$. What do $g(x)$ and $h(x)$ look like mod $p$?
What would this imply about $a_0$?

\prob
Prove that $f(x) = x^{p-1}+\ldots+x+1$ is irreducible when $p$ is prime:
\item{\bf(a)} Show that $(x-1)f(x) = x^p-1$.
\item{\bf(b)} Now set $x=y+1$, so $(x-1)f(x) = yf(y+1) = (y+1)^p-1$.
Study the binomial coefficients in the expansion of $(y+1)^p$, and apply
the Eisenstein criterion to $f(y+1)$.

\prob
Show that the following polynomials in $\ZZ[x]$ cannot be factored:
\item{\bf(a)} $x^3+6x^2+9x+12$
\item{\bf(b)} $x^2+x+6$

\prob
Let $A=\left[\matrix{1&2&3\cr 4&5&6\cr 8&8&8\cr}\right]$. Reduce $A$ to
diagonal form, using row and column operations.

\prob
Let $G$ be the {\it Abelian\/} group $G=\idealdef{a,b,c}{a^2b^2c^2 =
a^2b^2 = a^2c^2 = 1}$. Express $G$ as a product of free and cyclic groups.

\prob
Let $G$ be the {\it Abelian\/} group $G=\idealdef{a,b,c}{b^2c^2 =
a^6b^2c^2 = a^6b^4c^4 = 1}$. Express $G$ as a product of free and cyclic groups.

\prob
Let $R=k[x_1,\ldots,x_n]$ be the polynomial ring in $x_1,\ldots, x_n$
over a field $k$, and let $f_1,\ldots,f_m$ be $m$ polynomials in $R$. Let
$R^m$ be the free $R$-module $R^m = \setdef{(g_1,\ldots,g_m)}{g_i \in R
\;\hbox{for}\; 1\le i \le m}$. Let $M\subset R^m$ be the subset of {\it
syzygies\/} $M=\setdef{(g_1,\ldots,g_m)}{g_1 f_1 + \ldots + g_m f_m = 0}$.
\item{\bf(a)} Show that $M$ is an $R$-module.
\item{\bf(b)} Let $R=\QQ[x,y]$, $m=3$, and $f_1=x^2$,  $f_2=xy$,
$f_3=y^2$. Find a set of generators for $M\subset R^3$.

\prob
 Show that every element of $\FF_{25}$ is a root of the polynomial
$x^{25}-x$.

\prob
What is the minimal polynomial of $\alpha=\sqrt{2}+\sqrt{3}$ over
$\QQ$?

\bigskip

\centerline{This second set of problems are new, but are predicted by our
assignments.}

\bigskip

\prob
Let $F$ be a field, and let $f(x)$ be a polynomial of degree $n$ with
coefficients in $F$. Prove that $f(x)$ has at most $n$ roots in $F$.

\prob
Let $f(x)=a_n x^n + \ldots + a_1 x + a_0 \in \ZZ[x]$ be an integer polynomial,
and let $p$ be a prime integer which doesn't divide $a_n$. Prove that if the
remainder $\overline{f(x)}$ of $f(x)$ mod $p$ is irreducible, then $f(x)$ is
irreducible.

\prob
Let $F$ be a field of characteristic $\ne 2$, and let $K$ be an extension of
$F$ of degree 2. Prove that $K$ can be obtained by adjoining a square root:
$K=F(\delta)$, where $\delta^2=D$ is an element of $F$.

\prob
Let $F\subset K$ be a finite extension of fields. Define the degree symbol
$[K:F]$.

\prob
Let $F\subset K\subset L$ be a tower of finite field extensions. Prove
that $[L:F]=[L:K][K:F]$.

\prob
Consider the module $M=F[x]/((x-2)^3)$ over the ring $R=F[x]$ for a field $F$.
\item{\bf(a)} What is the dimension of $M$ as an $F$-vector space?
\item{\bf(b)} Find a basis for $M$ as an $F$-vector space, for which the matrix
representing multiplication by $x$ is in Jordan canonical form.

\bye