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\centerline{MATH 392 -- Seminar in Computational Commutative Algebra}
\centerline{Problem Set 3}
\centerline{{\bf Due:}  February 15, 2019}
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\noindent
1.  In the ring $\mathbb{Q}[x]$, 
\begin{enumerate}
\item[(a)] Use the \emph{Euclidean algorithm} for polynomials to compute
the greatest common divisor of $f_1(x) = x^3 + 3x^2 - 2$ and $f_2(x) = x^4 - 1$.
\item[(b)]  Find polynomials $A(x)$ and $B(x)$ such that 
$$\gcd(f_1(x),f_2(x)) = A(x)\cdot f_1(x) + B(x)\cdot f_2(x).$$
\item[(c)] Is $f(x) = x^6 + 7x + 1$ in the ideal $\langle f_1(x),f_2(x)\rangle$?
\end{enumerate}
\noindent
2.  In class we mentioned that there is a big analogy between the algebra of the ring of integers
$\mathbb{Z}$ and algebra of the polynomial ring $k[x]$ in one variable over a field $k$ because we have
similar division algorithms in both.  What should the analog of a \emph{prime number} $p \in \mathbb{Z}$
be in the polynomial ring $\mathbb{Q}[x]$?  Try to give a precise definition.  One thing to be 
careful of:  Nonzero constants $m/n \in \mathbb{Q}$ are also elements of $\mathbb{Q}[x]$ but they
have multiplicative inverses:  $(m/n)\cdot (n/m) = 1$.  This says you can always factor \emph{any}
nonzero constant out of the coefficients of a polynomial, for instance:
$$3x^3 + 4x + 1 = 3\cdot\left(x^3 + \frac{4}{3} x + \frac{1}{3}\right) = \frac{1}{7}\cdot \left(21 x^3 + 28 x + 7\right),$$
etc.  But those kinds of factorization are not the ``interesting'' ones.  An ``interesting'' factorization
would be something like 
$$x^2 - 1 = (x - 1)(x + 1).$$  
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\noindent
From the text:  Chapter 2, section 2/2, 7, 10, 11; Chapter 2, section 3/1, 9
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