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\centerline{\textbf{Dartmouth College}} \centerline{Mathematics 81/111
  --- Homework 3}

\begin{questions}

\question Given two relatively prime polynomials in $\C[x,y]$,
  B\'ezout's theorem in algebraic geometry gives the product of their
  degrees as an upper bound for the number of points of intersection.
  For example in $\R^2$ we expect the parabola $y-x^2=0$ will
  intersect a circle $x^2 + (y-a)^2 -4=0$ in anywhere from 0-4 points
  (but no more than 4), depending on the value of $a$.

  In this problem we give a first approximation to B\'ezout's theorem
  by proving that given two relatively prime polynomials $f(x,y)$,
  $g(x,y)$ in $k[x,y]$ ($k$ a field), their zero sets (i.e., the set
  of points $(x,y)$ where $f(x,y)=0=g(x,y)$) is finite.

  \begin{parts}
  \part Let $A$ be a UFD with field of fractions $K$.  Let
    $f, g \in A[x]$.  Show that $f,g$ are relatively prime in $A[x]$
    if and only if $f, g$ are relatively prime in $K[x]$ and their
    contents, $C(f)$ and $C(g)$, are relatively prime in $A$.

  \part Let $k$ be a field and $f,g \in k[x,y]$ be relatively
    prime. Show that the set of points $(x,y)$ where
    $f(x,y)=0=g(x,y)$) is finite.
  \end{parts}




\question Let $n \ge 1$ and consider the polynomial
  $f(x,y) = y^n - (x^3 -x) \in \Q[x,y]$.
  \begin{parts}
  \part Characterize the quotient $\Q[x,y]/(y,f(x,y))$ in terms of
    familiar rings.

  \part Show that $A= \Q[x,y]/(f)$ is an integral domain containing an
    isomorphic copy of $\Q[x]$. Show moreover that $A$ is finitely
    generated $\Q[x]$-module, in particular, that
    $A \subseteq \Q[x](1+(f)) + \Q[x](y+(f)) + \cdots +
    \Q[x](y^{n-1}+(f))$.
  

  \part Show that $B = \Q[x,y]/(x, f(x,y))$ is an integral domain if
    and only if $n=1$, but in any case is a finite dimensional vector
    space over $\Q$.

  \end{parts}



\question Let $A$ be a commutative ring with identity having prime
  characteristic $p$, and let \mbox{$\varphi: A \to A$} be the map
  $\varphi(a) = a^p$. The map $\varphi$ is called the
  \textit{Frobenius map}.

  \begin{parts}
  \part Show that $\varphi$ is a ring homomorphism.
  \part Henceforth assume that the ring $A$ is specialized to a field
    $F$ (having characteristic $p$).  Show that $\varphi: F \to F$ is
    injective.

  \part Show that if $F$ is a finite field, then $\varphi$ is
    surjective.
  \part Show that if $F$ is a finite field, then for any $f \in F[x]$,
    there exists a $g\in F[x]$ with $f(x^p) = (g(x))^p$.
  \part Show that if $F = \Z/p\Z$, then $\varphi$ is the identity map,
    and that for any $f \in F[x]$, $f(x^p) = (f(x))^p$.

  \end{parts}
  \vfill \rightline{(continued on next page)}
  \newpage

\question Hint: If you don't know where to begin, I would suggest
  looking up diagonalizable in Lang's index.  The reference will give
  (as a homework problem) an important theorem.

  \begin{parts}
  \part Let $A$ be an $n\times n$ matrix over the complex numbers,
    $\C$, for which $A^k = I_n$ is the identity matrix for some
    integer $k \ge 1$.  Show that $A$ is diagonalizable.

  \part Let $F$ be a field of prime characteristic $p$, and $A =
    \begin{pmatrix} 1&\alpha\\0&1\end{pmatrix} \in M_2(F).$ Show that
    $A^p = I_2$, and that $A$ is diagonalizable if and only if
    $\alpha = 0$.

  \end{parts}

\question We have seen a number of tests to help determine whether a
  polynomial is irreducible, and they will serve you well.  On the
  other hand, neither one size nor one tool fits all problems.

  \begin{parts}
  \part Show that for all $n \ge 1$, $f(x) = (x-1)(x-2)\cdots(x-n) -1$
    is irreducible in $\Z[x]$.

  \part Show that for all $n \ge 1$ (except $n=4$),
    $f(x) = (x-1)(x-2)\cdots(x-n) +1$ is irreducible in $\Z[x]$.  This
    part requires a bit more persistence than the first part.
  \end{parts}



\end{questions}

\end{document}







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