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\begin{flushright}
  Your Name Goes Here
\end{flushright}
\vspace{.5in}



\centerline{\textbf{Dartmouth College}} \centerline{Mathematics 81/111
  --- Homework 5}

\begin{questions}

\question Let $L/K$ be an extension of fields.  We say that
  \textbf{$K$ is algebraically closed in $L$} if the only elements of
  $L$ which are algebraic over $K$ are the elements of $K$.  Let $x$
  be transcendental over $K$.
  \begin{parts}
  \part Show that $K$ is algebraically closed in $K(x)$.
  \part Let $\xi$ be any element of $K(x) \setminus K$.  Show that
    $K(x)/K(\xi)$ is an algebraic extension.
  \end{parts}

\question Let $K/F$ be a field extension of degree $n$.
  \begin{parts}
  \part For any $\alpha \in K$, show that $T_\alpha: K \to K$ given by
    left multiplication by $\alpha$ is an $F$-linear transformation on
    $K$.



  \part Prove that $K$ is isomorphic to a subfield of $M_n(F)$.  In
    particular this shows that $M_n(F)$ contains an isomorphic copy of
    all field extensions of $F$ having degree dividing $n$.
  \end{parts}

\question Finding minimal polynomials.
  \begin{parts}
  \part Let $L/F$ be an algebraic extension of fields, and
    $\alpha \in L$.  Let
    $m_{\alpha,F} = x^n + a_{n-1}x^{n-1} + \cdots + a_0$ be the
    minimal polynomial of $\alpha$ over $F$, and put $K = F(\alpha)$.
    Consider the linear transformation $T_\alpha$ of the previous
    problem.  Compute the matrix $[T_\alpha]_\B$ of $T_\alpha$ with
    respect to the basis $\B = \{1, \alpha, \dots, \alpha^{n-1}\}$ of
    $K/F$, and show its characteristic polynomial is $m_{\alpha,F}$.

  \part First, let's do a simple example.  Let $F = \Q$ and
    $\alpha = \root 5\of 2$. Show that the characteristic polynomial
    of $[T_\alpha]_\B$ is the expected $x^5 - 2$.

  \part Now suppose that
    $\beta = 1 + \root 5 \of 8 + \root 5\of {16}$.  We know from a
    simple argument on degrees of towers that the degree of $\beta$
    over $\Q$ is 5. Computing the matrix of $T_\beta$ with respect to
    $\{1, \beta, \dots, \beta^4\}$ would be painful.  Instead, compute
    the matrix of $T_\beta$ with respect to the basis
    $\B = \{1, \alpha, \dots, \alpha^4\}$ where
    $\alpha = \root 5 \of 2$ and the resulting characteristic
    polynomial.  Is this $m_{\beta,\Q}$?  If so why?  If not, why not?
  \end{parts}

\question 

  \begin{parts}
  \part Let $K$ be a field of characteristic $p$, and let $a \in K$.
    Show that if $a$ has no $p$th root in $K$, then $x^{p^n} -a $ is
    irreducible in $K[x]$ for any $n \ge 1$.

  \part Let $\root7\of 2$ be the real seventh root of 2, and
  $\zeta_7$ a primitive 7th root of unity in $\C$. Show that $-1$
  cannot be written as a sum of squares in $\Q(\root7\of2 \zeta_7)$


  
  \end{parts}




\question A field $K$ is called \textbf{perfect} if either it has
  characteristic 0, or has characteristic $p$ and $K^p = K$ (that is,
  the Frobenius map is an automorphism of $K$).

  \begin{parts}
  \part Show that any algebraic extension of a perfect field is
    separable.
  \part Give (and justify) an example of a field which is not perfect.
  \end{parts}


\end{questions}

\end{document}







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