\magnification=\magstep1
\centerline{MATH 392 -- Seminar in Computational Commutative Algebra}
\centerline{Midterm Exam}
\centerline{March 22, 2019}
\bigskip
\noindent
{\it Directions}
\bigskip
Do all work in the blue exam book.  Do not place anything you want
considered for credit on this sheet.  There are 100 possible points,
distributed as indicated in the questions.
\vskip 40pt
\noindent
I.
\item{A)} (20) Prove that every ideal $I$ in the polynomial ring $k[x]$ (one variable)
is principal (that is, $I = \langle g(x)\rangle$ for some single polynomial).
\medskip
\item{B)} (10) Find $g(x)$ as in part A for the ideal $I = \langle x^2 + 7x + 10, x^3 + x^2 + 4\rangle$
in ${\bf Q}[x]$.
\bigskip
\noindent
II.  
\item{A)} (15) Define:  $G$ is a Gr\"obner basis for an ideal $I \subseteq k[x_1,\ldots,x_n]$ 
with respect to a monomial order $>$.
\medskip
\item{B)} (10) Assuming the statement of Dickson's Lemma, prove that Gr\"obner bases exist for every ideal $I$
in $k[x_1,\ldots,x_n]$ and with respect to every monomial order. 
\medskip
\item{C)} (5) What else do you need to know in order for the result from part B to give a proof
of the Hilbert Basis Theorem?  (You don't need to give the proof, just say what else must be proved.) 
\bigskip 
\noindent
III.  
\item{A)} (20)  State and prove the Elimination Theorem.  
\medskip
\item{B)} (10)  A certain ideal $J \subset {\bf Q}[x,y,z]$ has a Gr\"obner basis
$$B = \{x^3 - 3x^2 + 2x, x^2 y - xy, y^2 - y, z - xy\}$$
with respect to the lexicographic order with $z > y > x$.   What are bases for 
the elimination ideals 
$$J_1 = J \cap {\bf Q}[y,x] \quad {\rm and} \quad J_2 = J\cap {\bf Q}[x]?$$
\item{C)} (10)  Use the information in part B to determine all of the points in $V(J)$.
\end
 
