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\begin{document}

\noindent
{\large \bf Old Prelim Problems.}

\bigskip

\begin{enumerate}
\item[(1)] %S00 
Find the cardinality of the set of all subrings of $\BQ$, 
the field of rational numbers.

\bigskip
\item[(2)] %F98  
Let $\m r$ be a finite ring with identity. Let $a$ 
be an element of $\m r$
which is not a zero divisor. Show that $a$ is invertible.

\bigskip
\item[(3)] %S98  
Let $\m a$ be the ring of real $2\times 2$ upper triangular
matrices. What
are the $2$-sided ideals in $\m a$? Justify your answer.

\bigskip
\item[(4)] %F96  
Let ${\BZ}[x]$ be the ring 
of polynomials in the indeterminate $x$ with
coefficients in the ring $\BZ$ of integers. 
Let $I\subseteq {\BZ}[x]$ be the ideal generated by
$13$ and $x-4$. Find an integer $m$ such that 
$0\leq m \leq 12$ and
$(x^{26} + x + 1)^{73} - m \in I$.

\bigskip
\item[(5)] %S96  
Prove that $f(x) = x^4 + x^3 + x^2 + 6x + 1$ is irreducible over 
$\BQ$.

\bigskip
\item[(6)] %F95  
Prove that ${\BQ}[x, y]/(x^2 +y^2-1)$
is an integral domain and that its 
field of fractions is isomorphic to the field of rational functions
${\BQ}(t)$.


\bigskip
\item[(7)] %F95  
Let $f(x) \in {\BQ}[x]$ 
be a polynomial with rational coefficients.
Show that there is a $g(x) \in {\BQ}[x]$, 
$g \neq 0$, such that $f(x)g(x) = a_2x^2 +a_3x^3 +
a_5x^5 + \cdots + a_px^p$ 
is a polynomial in which only prime exponents appear.

\bigskip
\item[(8)] %S95 
Suppose that $\m r$ is a subring of a commutative ring 
$\m s$ and that the additive subgroup of
$\m r$ is of finite index $n$ in $\m s$. 
Let $m$ be an integer that is relatively prime to $n$.
Prove that the natural map 
$\m r/mR \rightarrow \m s/mS$ is a ring isomorphism.

\bigskip
\item[(9)] %F94  
Let $\m r$ be a ring with identity, and let 
$u$ be an element of $\m r$
with a right inverse. 
Prove that the following conditions on $u$ are equivalent:
\begin{enumerate}
\item[(i)]
$u$ has more than one right inverse;
\item[(ii)]
$u$ is a zero divisor;
\item[(iii)]
$u$ is not a unit.
\end{enumerate}

\bigskip
\item[(10)] %S94 
Find all automorphisms of ${\BZ}[x]$, the ring of polynomials over
${\BZ}$.


\bigskip
\item[(11)] %S93 
Prove that no commutative ring with identity has additive group
isomorphic to ${\BQ}/{\BZ}$.

\bigskip
\item[(12)] %F92  
Let $I$ denote the ideal in ${\BZ}[x]$, 
the ring of polynomials with coefficients
in ${\BZ}$, generated by $x^3 + x + 1$ and $5$. 
Is $I$ a prime ideal?

\bigskip
\item[(13)] %F90  
Let $\m r$ be a ring with $1$, 
and let $I$ be the left ideal of $\m r$ generated
by $\{ ab-ba \;|\; a, b \in R \}$. 
Prove that $I$ is a two-sided ideal.

\bigskip
\item[(14)] %F90  
Prove that $\sqrt{2} + \sqrt[3]{3}$
is irrational.

\bigskip

\item[(15)] %S89 
Let $\m r$ be a ring with at least two elements. Suppose that for
each nonzero $a$ in $\m r$ 
there is a unique $b$ in $\m r$ (depending on $a$) with 
$aba = a$.
Show that $\m r$ is a division ring.


\bigskip

\item[(16)] %S89. 
Let $\m r$ be a commutative ring with $1$ 
containing an element
$a$ with $a^3 = a +1$. Further, let $I$ 
be an ideal of $\m r$ of index $< 5$ in
$\m r$. Prove that $I = R$.
Show that there exists a commutative ring with $1$ 
that has an element
$a$ with $a^3 = a + 1$ 
and that contains an ideal of index $5$.

\bigskip

\item[(17)] %S88 
Let $\m r$ be a commutative ring with unit element and 
$a\in R$. Let $n$ and $m$ be positive integers, 
and write $d = \gcd(n, m)$. Prove that the ideal
of $\m r$ generated by $a^n-1$ and $a^m-1$
is the same as the ideal generated by
$a^d-1$. 

\bigskip

\item[(18)] %F86  
Show that the polynomial $p(z) = z^5 - 6z + 3$ 
is irreducible and has five distinct
complex roots, of which exactly three (and not five) are real.

\bigskip

\item[(19)] %S86 
Let $\BZ$ be the ring of integers, $p$ a prime, and 
${\BF}_p = {\BZ}/p{\BZ}$ the
field of $p$ elements. 
Let $x$ be an indeterminate, and set 
$\m r_1 = {\BF}_p[x]/(x^2-2)$, $\m r_2 = {\BF}_p[x]/(x^2-3)$.
Determine whether the rings $\m r_1$ and 
$\m r_2$ are isomorphic
in each of the cases $p = 2, 5, 11$.

\bigskip

\item[(20)] %S85 
Let $\omega = e^{2\pi i/7}$
be a primitive $7$-th root of unity. Find a cubic
polynomial with integer coefficients having 
$\zeta = \omega + \omega^{-1}$ as a root.

\bigskip


\item[(21)] %F84 
Let $a$ be an element in a field $\BF$ and let 
$p$ be a prime. Assume
$a$ is not a $p$-th power. Show that the polynomial 
$x^p-a$ is irreducible in ${\BF}[x]$.

\bigskip


\item[(22)] %S84 
Let $\m r$ be a principal ideal domain 
and let $I$ and $J$ be nonzero
ideals in $\m r$. Show that $IJ = I\cap J$ 
if and only if $I + J = R$.
\end{enumerate}
\end{document}


