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\centerline{ \bigbf More Problems, September 18, 1989 }
\bs
\centerline { \bigbf Math 401 -- Second Course in Abstract Algebra, I}
\ms
\centerline { \bigbf Prof. Daniel Grayson }
\ms
\centerline { 2:00 PM MWF, Section F1, 241 Altgeld Hall }
\bs

A big hurdle for a graduate student is the transition from course-taking to
thesis-research, and my goal is not to force you to make that transition now,
but rather to make it easier for you when it eventually occurs.  That is why we
are working on these problems, and why the problems are phrased so vaguely.  I
intend that everyone have a comfortable and leisurely exposure to this sort of
problem-solving activity, so please don't get distraught if you seem to be more
of a spectator than a participant in your group -- you are still at an early
stage in your mathematical careers.  But a spectator should be an active
spectator: ponder at home the progress the groups made in class, and try to
reproduce it, to record it accurately, and to understand what motivates each
step.

\bs
\problem 5

Let $\omega$ be a primitive n-th root of $1$ in some field.  Let $A$ be the
n-by-n matrix whose entries are $A_{ij} = \omega^{ij}$, where $0 \le i,j \le
n-1$.  Find a simple expression for the inverse matrix $A^{-1}$.  Find an even
simpler proof for the invertibility of $A$.

\bs
\problem 6

Consider an automorphism $\sigma$ of a field $K$ whose order $n$ is finite.
Let $F = \{ x \in K \,|\, \sigma(x) = x \}$ be the fixed field of $\sigma$.
Study the structure of the field extension $F \subseteq K$, using the ideas
discovered in treating the roots of the general cubic equation.  Make
assumptions which enable you to prove something.


\vfill

\noindent
{\bf Homework:} Read sections III.6, and IV.1.  Do the following
problems, but don't hand in the parenthesized ones unless you feel uncertain
about your solution: 
III.6, Exercises (1), (3), 5, 6, (7), 8, (10), (11), (12), (15), (16);
IV.1, Exercises (1), (2), (3), 4, 5, (7), 9, 12, (13), 15.  
Turn in on Wednesday, September 27.

\vfill
\eject


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