\chapter{Polynomials in one variable over a field}

 


In the following, 
   F is a field and F[x] is the ring of polynomials over F



\begin{theorem} F[x] is an integral domain.
\end{theorem}


\begin{theorem}  The units of F[x] are the nonzero constant polynomials in F[x]
\end{theorem}


\begin{theorem} The division algorithm holds:  For each a,b in F[x] with $b <> 0$, 
       there are  unique $q,r$ in F[x] such that  $a = bq + r$ where $r = 0$ or
       $deg r < deg b$.
\end{theorem}
       

\begin{theorem} The divisor order on F[x] is defined by  $f | g$ if and only if  $ g = fq$
      for some q in F[x].   It has the following properties:
\begin{enumerate}
      \item  $f | f$ for each f in F[x]
      \item    If $f | g$ and $g | h$ then  $f | h$.
        \item    If $f | g$ and $g | f$ then  $f = g = 0$ or  $f = ug$ for some unit u.  
       Note: Two polynomials who divide each other are called associates.
\end{enumerate}
\end{theorem}
  

\begin{theorem}If  f,g are in F[x] with $fg <> 0$, then  gcd(f,g) is defined as the monic    
     polynomial 
of  largest degree which divides both f and g.   
\begin{enumerate}
       \item gcd(f,g) is unique.
      \item  The Euclidean Algorithm:   If  $deg f <= deg g$, then use div algo
        to write $g = fq1 + r1$, then $f = r1 q2 + r2$, then $r1 = r2 q3  + r 4$,
        etc.   The last nonzero remainder is a unit times the gcd of f and g.
        This algorithm extends to finding solutions to  $f x + g y = gcd(f,g)$.
\end{enumerate}
\end{theorem}


\begin{definition} A polynomial f in F[x] of degree greater than 0 is called irreducible
       in F[x] if it cannot be factored into two polynomials of smaller degree.
       Equivalently, its only divisors are its associates and the units.
\end{definition}



\begin{theorem}[Unique factorization Theorem]
       Every nonconstant polynomial f in F[x] factors into
       a product of powers of irreducible polynomials in F[x].  This
       factorization is unique up to associates and the order in which 
       the terms are written down.
\end{theorem}



\begin{theorem}[Factor  Theorem]  
      For each  f in F[x] and each a in F,  $x - a |  f$   if  and only if  $f(a) = 0$.
\end{theorem}


\begin{theorem}[Number of roots theorem] 
       If  f is a polynomial of degree n  in F[x], then $f(x) = 0$ has at most n    
    solutions in F.
\end{theorem}

\begin{theorem}[Fundamental theorem of Algebra]   
       Every   polynomial with coefficients in C, the field of complex
       numbers, has exactly n roots in C, counting multiplicities.
\end{theorem}



   

