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\title{Math 129: Algebraic Number Theory\\
{\bf Lecture 18: Independence, Finite Residue Field Case}}
\author{William Stein}
\date{Tuesday, April 13, 2004}
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\begin{align*}
   b_r &= \frac{c_1^r}{1+c_1^r} \cdot a_1 + 
     \frac{c_2^r}{1+c_2^r} \cdot a_2\\
     &=\frac{2^r}{1+2^r} \cdot (-1) + 
     \frac{(1/2)^r}{1+(1/2)^r} \cdot 8.
   \end{align*}
We have $b_1 = 2$, $b_2 = 4/5$, $b_3 = 0$, $b_4 = -8/17$, 
$b_5 = -8/11$, $b_6 = -56/55$.  None of the $b_i$ work for $i<6$,
but $b_6$ works.
\end{example} 

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\section{Finite Residue Field Case}



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