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\begin{document}

\noindent{\bf MATHEMATICS} {\bf 362} \hspace{2in} {\bf Name}
\hrulefill

\noindent{\bf Test 1}

\noindent February 7, 2001 \hfill 4 questions

\next
\parindent=0mm
{\sl In questions 2, 3, 4, $\Q$ denotes the field of rational
numbers.}

\vspace{4mm}
 {\bf 1}. ({\sl 25 points})

\vspace{3mm}
\noindent (a) Define what it means for a field extension $L:K$
to be a finite extension, and define the degree $[L:K]$.

\vspace{1.25in}

 (b) Let $K,L,M$ be fields, with $K \subseteq L \subseteq M$,
and suppose $M:L$ and $L:K$ are finite extensions. Explain
carefully how to obtain a basis $B$ of $M$ over $K$, from a
basis of $M$ over $L$ and a basis of $L$ over $K$. ({\sl Proof
that $B$ is actually a basis is not required.})

\vspace{1.5in}

 (c) Suppose $K,L,M,N$ are fields, with $K \subseteq L \subseteq
M \subseteq N$, and suppose that $[M:K]=15$, $[N:L]=6$. Show
that $[N:K]$ must be either 30 or 90 .


\newpage

{\bf 2}. ({\sl 25 points}) Let $L:K$ be a field extension,
$\alpha
\in L$.

\vspace{2mm}

(a) Explain what is meant by $K(\alpha)$.

\vspace{1in}

 (b)  If $f$ is an irreducible polynomial of degree $n$ in
$K[t]$, and $\alpha$ is a root of $f$, write down a basis for
$K(\alpha)$ over $K$. ({\sl Proof not needed.})

\vspace{1in}
(c) Show that the polynomial $g=t^2+t+1$ is irreducible over the
field $\Q$.

\vspace{2.4in}
 (d) Suppose the complex number $\alpha$ is a root of the
polynomial $g$ of (c) above. If $a,b,c,d\in \Q$, express the
product $(a+b\alpha)(c+d\alpha)$ in the form
$e+f\alpha$, where $e,f\in \Q$.



\newpage

{\bf 3}. ({\sl 25 points})
\vspace{4mm}

(a) If $\alpha,\beta$ are both algebraic over a field $K$, give
a condition involving polynomials under which there exists a
$K$-isomorphism (i.e., an isomorphism leaving every element of
$K$ fixed)
$$\phi : K(\alpha) \rightarrow K(\beta),$$ such that
$\phi(\alpha)=\beta$. ({\sl No justification
needed.})

\vspace{1in}

For the next two parts of the question, let $\alpha$ be as in
Question 2(d), i.e.,
$\alpha$ is a complex root of the polynomial $g=t^2+t+1$.
\vspace{3mm}
\newline (b) Show that there exists
a
$\Q$-automorphism $$\phi:\Q(\alpha)\rightarrow \Q(\alpha),$$ such
that $\phi(\alpha)=-1-\alpha$.

\vspace{1.8in}

(c) If $\gamma=a+b\alpha$, where $a,b \in \Q$, compute the
product $\gamma \cdot \phi(\gamma)$, and find a formula for
$(a+b\alpha)^{-1}$.




\newpage

{\bf 4}. ({\sl 25 points}) \vspace{2mm}

(a) Give a careful statement ({\sl without proof}) of a theorem
connecting field extensions with geometric constructibility.

\vspace{1.3in}

(b) It is easy to bisect angles by means of straightedge and
compass. Given a number $\theta$, let $\alpha = \cos \theta,
\beta = \cos 2\theta$. By using a suitable trigonometric
identity, show that $$\Q(\beta)\subseteq \Q(\alpha),$$
$$[\Q(\alpha):\Q(\beta)] \leq 2.$$

\vspace{2in}

(c) If
$\alpha _{n} = \cos(\pi/2^{n})$, where
$n$ is a positive integer, show that
$$[\Q(\alpha _{n}):\Q]=2^m,$$ where $m < n$.

\end{document}


