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%(NZM, Problem ) 

\ctln{\bf Math 445 Homework 6}

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\ctln{Due Friday, October 17}

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\item{21.} Show that if an integer $n$ can be expressed as the sum of the squares of two {\it rational} numbers

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\ctln{$\displaystyle n = ({{a}\over{b}})^2 + ({{c}\over{d}})^2$ ,}

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\item{} then $n$ can be expressed as the sum of the squares of two {\it integers}.

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\item{} (Hint: Not directly! Show that $n$ has the correct prime factorization....)

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\item{22.} [NZM, p. 106, \#\ 2.8.8] Determine how many solutions (mod 17) each of the following 
congruence equations has:

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\hskip1in (a) $x^{12}\equiv 16$ (mod 17) \hskip1in (b) $x^{48}\equiv 9$ (mod 17)

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\hskip1in (c) $x^{20}\equiv 13$ (mod 17) \hskip1in (d) $x^{11}\equiv 9$ (mod 17)


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\item{23.}  If $p$ is a prime, and $p\equiv 3$ (mod 4), show that the congruence equation

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\item{} $x^4\equiv a$ (mod $p$) has a solution $\Leftrightarrow$ $x^2\equiv a$ (mod $p$) does.

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\item{} On the other hand, show (by example) that if $p\equiv 1$ (mod 4) this result need \underbar{not}
be true.


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\item{24.} [NZM, p.106, \#\ 2.8.13] Show that, for a prime $p$, the numbers $1^k,2^k,\ldots (p-1)^k$
are all \underbar{distinct} mod $p$ $\Leftrightarrow$ $(k,p-1)=1$.


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