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{{\Large\bf MODERN ALGEBRA I}\\
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{\Large\bf HOMEWORK ASSIGNMENT IV}\\
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{\bf (Turn in underlined problems.)}}
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{\bf 
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\underline{\large\bf CHAPTER}\makebox[2in]{}\=\underline{\large\bf PROBLEMS}\\

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1 \> \underline{1.50}, \underline{1.56},
\underline{1.57}
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\noindent
\underline{\large\bf ADDITIONAL PROBLEMS}
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{\bf 

\underline{1.} For which $0\leq a < 30$ is 
$ax\equiv 1\;({\rm mod}\;30)$ solvable?
(If this congruence is solvable for some
$a$ we say that $a$ is {\em invertible modulo $30$}.)

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\underline{2.}
Let $p$ be an odd prime.
Show that
\[1^p + 2^p + 3^p + \cdots + p^p\equiv 0\;({\rm mod}\;p).\]

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3. An integer $q\geq 2$ is an 
{\em absolute pseudoprime} if
$a^q\equiv a\;({\rm mod}\;q)$ holds for all $a$.
We have shown that every prime is an
absolute psuedoprime,
but some other numbers (such as $n=561$) are 
absolute pseudoprimes.
Show that if $q$ is an absolute pseudoprime, then $q$
is squarefree. 

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