\vsize 9in
\hsize 6.5in
\def\goodbreak{\par\penalty-500 }
\def\problem#1\par{\bigskip\bigskip\goodbreak\centerline{Problem #1}\bigskip}
\font\bigbf=cmbx12
\def\ni{\noindent}
\def\bs{\bigskip}
\def\ms{\medskip}
\def\ss{\smallskip}
\def\subhead{\bf}
\centerline{ \bigbf Problems, August 25, 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
\problem1

Here are some factorizations generated by the computer.  The factors on the
right hand side are irreducible, or at least, are claimed to be so by the computer.
Examine them, looking for patterns, and try to explain those patterns.

$$
\eqalign{
t - 1 &= t-1 \cr
{t^2} - 1 &= 
   \left( t-1 \right)  
   \left( 1 + t \right)  \cr
{t^3} - 1 &= 
   \left( t-1 \right)  
   \left( 1 + t + {t^2} \right)  \cr
{t^4} - 1 &= 
   \left( t-1 \right)  
   \left( 1 + t \right)   
   \left( 1 + {t^2} \right)  \cr
{t^5} - 1 &= 
   \left( t-1 \right)   
   \left( 1 + t + {t^2} + {t^3} + {t^4} \right)  \cr
{t^6} - 1 &= 
   \left( t-1 \right)  
   \left( 1 + t \right)   
   \left( 1 - t + {t^2} \right)  
   \left( 1 + t + {t^2} \right)  \cr
{t^7} - 1 &= 
   \left( t-1 \right)   
   \left( 1 + t + {t^2} + {t^3} + {t^4} + {t^5} + {t^6} \right)  \cr
{t^8} - 1 &= 
   \left( t-1 \right)  
   \left( 1 + t \right)   
   \left( 1 + {t^2} \right)  
   \left( 1 + {t^4} \right)  \cr
{t^9} - 1 &= 
   \left( t-1 \right)  
   \left( 1 + t + {t^2} \right)   
   \left( 1 + {t^3} + {t^6} \right)  \cr
{t^{10}} - 1 &= 
   \left( t-1 \right)  
   \left( 1 + t \right)   
   \left( 1 - t + {t^2} - {t^3} + {t^4} \right)   
   \left( 1 + t + {t^2} + {t^3} + {t^4} \right)  \cr
{t^{11}} - 1 &= 
   \left( t-1 \right)   
   \left( 1 + t + {t^2} + {t^3} + {t^4} + {t^5} + {t^6} + {t^7} + {t^8} + 
          {t^9} + {t^{10}} \right)  \cr
{t^{12}} - 1 &= 
   \left( t-1 \right)  
   \left( 1 + t \right)   
   \left( 1 + {t^2} \right)  
   \left( 1 - t + {t^2} \right)   
   \left( 1 + t + {t^2} \right)  
   \left( 1 - {t^2} + {t^4} \right)  \cr
{t^{13}} - 1 &= 
   \left( t-1 \right)   
   \left( 1 + t + {t^2} + {t^3} + {t^4} + {t^5} + {t^6} + {t^7} + {t^8} + {t^9}
        + {t^{10}} + {t^{11}} + {t^{12}} \right)  \cr
{t^{14}} - 1 &= 
   \left( t-1 \right)  
   \left( 1 + t \right)   
   \left( 1 - t + {t^2} - {t^3} + {t^4} - {t^5} + {t^6} \right)   
   \left( 1 + t + {t^2} + {t^3} + {t^4} + {t^5} + {t^6} \right)  \cr
{t^{15}} - 1 &= 
   \left( t-1 \right)  
   \left( 1 + t + {t^2} \right)   
   \left( 1 + t + {t^2} + {t^3} + {t^4} \right)   
   \left( 1 - t + {t^3} - {t^4} + {t^5} - {t^7} + {t^8} \right)  \cr
{t^{16}} - 1 &= 
   \left( t-1 \right)  
   \left( 1 + t \right)   
   \left( 1 + {t^2} \right)  
   \left( 1 + {t^4} \right)   
   \left( 1 + {t^8} \right)  \cr
{t^{17}} - 1 &= 
   \left( t-1 \right)   
   \left( 1 + t + {t^2} + {t^3} + {t^4} + {t^5} + {t^6} + {t^7} + {t^8} + 
     {t^9} + {t^{10}} + {t^{11}} + {t^{12}} + {t^{13}} + {t^{14}} +    {t^{15}} + {t^{16}} \right)  \cr
{t^{18}} - 1 &= 
   \left( t-1 \right)  
   \left( 1 + t \right)   
   \left( 1 - t + {t^2} \right)  
   \left( 1 + t + {t^2} \right)   
   \left( 1 - {t^3} + {t^6} \right)  
   \left( 1 + {t^3} + {t^6} \right)  \cr
{t^{19}} - 1 &= 
   \left( t-1 \right)   
   \left( 1 + t + {t^2} + {t^3} + {t^4} + {t^5} + {t^6} + {t^7} + {t^8} +   
	 {t^9} + {t^{10}} + {t^{11}} + {t^{12}} \right. \cr
	& \hskip 2.5in \left.
	{} + {t^{13}} + {t^{14}} +   
         {t^{15}} + {t^{16}} + {t^{17}} + {t^{18}} \right)  \cr
{t^{20}} - 1 &= 
   \left( t-1 \right)  
   \left( 1 + t \right)   
   \left( 1 + {t^2} \right)  
   \left( 1 - t + {t^2} - {t^3} + {t^4} \right)   
   \left( 1 + t + {t^2} + {t^3} + {t^4} \right)   
   \left( 1 - {t^2} + {t^4} - {t^6} + {t^8} \right)  \cr
{t^{21}} - 1 &= 
   \left( t-1 \right)  
   \left( 1 + t + {t^2} \right)   
   \left( 1 + t + {t^2} + {t^3} + {t^4} + {t^5} + {t^6} \right)   
   \left( 1 - t + {t^3} - {t^4} + {t^6} - {t^8} + {t^9} - {t^{11}} + {t^{12}} \right)  \cr
}
$$

\problem 2

It seems plausible that as the coefficients of a polynomial vary, the roots
vary in a continuous fashion.  Here we take the coefficients to be
real numbers or complex numbers, so that the notion of continuity makes sense.
Decide whether this is is true.

\problem 3

Suppose a polynomial $f(x) = x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$ has roots
$\alpha_1, \dots, \alpha_n$.  Suppose also that $P(T_1,\dots,T_n)$ is a
polynomial with the property that for any permutation $\sigma$, we have 
$P(T_1,\dots,T_n) = P(T_{\sigma 1},\dots,T_{\sigma n})$.  Such a polynomial is
called symmetric.
Develop an algorithm for locating a polynomial $Q(X_0,\dots,X_{n-1})$ so that
$P(\alpha_1,\dots,\alpha_n) = Q(a_0,\dots,a_{n-1})$.

\problem 4

Try to find a formula for the roots of a cubic polynomial $x^3 + a x^2 + b x +
c$.

\end
