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\begin{document}

\textbf{Math 6320, Assignment 2  \hfill Due: Weekend of February 9}

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\begin{enumerate}

\item
\begin{enumerate}
\item
If a polynomial $f(x):=a_{0}+a_{1}x+\cdots a_{n}x^{n}$ has a rational root $p/q$ with $(p,q)=1$, then $p$ divides $a_{0}$ and $q$ divides $a_{n}$. Prove this assertion.
\item
Prove that a cubic polynomial in $\bbZ[x]$ with no rational roots is irreducible in $\bbQ[x]$. 
\item
Is the polynomial $x^{3}-4x^{2}-(9/2)x-(5/2)$ irreducible in $\bbQ[x]$?
\end{enumerate}

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\item 
Study the irreducibility of $x^{4}+1$ in $\bbQ[x]$ and factorize it as a product of irreducible factors in $\bbQ(\zeta)[x]$, where $\zeta$ is one of its roots.

%\textbf{Solution:} 

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\item
For each prime $p$ and integer $n$, verify that the polynomial $x^{n}+px+p^{2}$ is irreducible in $\bbQ[x]$.

%\textbf{Solution:} 

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\item
Give an example of an irreducible quadratic polynomial $f(x)$ in $\bbQ[x]$ such that $f(x^{2})$ is reducible.
%\textbf{Solution:} 

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\item
Let $F$ be a field and $f(x),g(x)$ polynomials in $F[x]$. Let $\alpha$ and $\beta$ be roots of $f(x)$ and $g(x)$, respectively, in some extension of $F$. 
\begin{enumerate}
\item
Verify that the polynomial $R(z):=\res_{x}(f(x), g(z-x))$ is a polynomial in $F[z]$ having $a+b$ as a root.
\item 
More generally, for any polynomial $p(x,y)$ in $F[x,y]$, the element $p(\alpha,\beta)$ is a root of the polynomial
\[
\res_{x}(f(x), \res_{y}(z-p(x,y),g(y))) \quad\text{in $F[z]$.}
\]
\item
Verify that $\alpha\beta$ is a root of the polynomial $\res_{x}(f(x),x^{n}g(z/x))$, where $n=\deg(g(x))$.
\item
For a polynomial $p(y)$ in $F[y]$, describe a polynomial over $F$ that vanishes at $p(\alpha)$.
\end{enumerate}
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\item
Let $\alpha$ be a root of the polynomial $f(x)=x^{3}+2x+2$ in $\bbQ[x]$. Since $f(x)$ is irreducible, by the Eisentein criterion, the
$\bbQ$-vector space  $\bbQ(\alpha)$ has a basis $1,\alpha,\alpha^{2}$.  Express the following elements in $\bbQ(\alpha)$ in terms of this basis:
\[
a^{-1}\,, \qquad 1/(a^{2}+a+1)\,, \qquad \text{and} \qquad a^{6}+3a^{4}+2a^{3}+6a
\]
Determine the minimal polynomials, over $\bbQ$, of these elements.

%\textbf{Solution:} 

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\item
Let $R$ be a ring and $f(x):=a_{0}+a_{1}x+\cdots a_{n}x^{n}$ a polynomial over $R$. Prove the following assertions.
\begin{enumerate}
\item
$f(x)$ is a unit in $R[x]$ if and only if $a_{0}$ is a unit and each $a_{i}$ is nilpotent.
\item
$f(x)$ is nilpotent if and only if each $a_{i}$ is nilpotent.
\item
$f(x)$ is a zerodivisor if and only if there exists a nonzero element $r\in R$ such that $r\cdot f(x)=0$.
\end{enumerate}

%\textbf{Solution:} 

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\end{enumerate}

\end{document}

