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\def\title{Homework 8, April 25}
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\bs  
A hint for exam problem 4 is to look at VII.2.11.

\bs
These problems are due Wednesday, May 2.

\bs
Do p.~442: 4 (see IX.6.9).

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\bs
The following exam question is to be done independently.
\bs

{\bf Exam Problem 5:}
See definition IX.6.1.  Let $F$ be a field with a primitive $n$-th root of
unity $\zeta$, and supose $a$ and $b$ are nonzero elements of $F$.  Let $A$ be the
$F$-algebra given by the generators $x$ and $y$ and the relations
$$
\align
 x^n&=a
\\
 y^n&=b
\\
x y &= \zeta y x
\endalign
$$
Show that $A$ is a central simple algebra over $F$ of dimension $n^2$.

\bs 

{\bf Exam Problem 6:} Show that the ring of $2$ by $2$ matrices over a field
$F$ of the form
$\pmatrix a & b \\ 0 & d \endpmatrix$ is not a semisimple ring.

\bs

{\bf Exam Problem 7:} Let $F$ be a finite field of characteristic $p$, and let
$G$ be the group of $n$ by $n$ invertible matrices over $F$.  Let $H$ be the
subgroup of $G$ consisting of those matrices which are $0$ below the diagonal
and $1$ on the diagonal, and have arbitrary elements above the diagonal.  (a)
Show that $H$ is a $p$-Sylow subgroup of $G$.  (b) Find out how many $p$-Sylow
subgroups $G$ has, and verify that the number is congruent to $1$ modulo $p$.

\bye


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