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\begin{document}

\begin{center}
{\bf Math 601 -- Spring 2026 -- Harry Tamvakis}
\end{center}


\begin{center}
{\bf PROBLEM SET 6 -- Due April 2, 2026}
\end{center}


\skipline
\noin {\bf A1)} Prove that a finitely generated abelian group $A$
is free if and only if $\Ext^1_\Z(A,\Z)=0$.



\skipline \noin {\bf A2)} Prove that if $0 \to M' \to M \to M'' \to 0$ is a 
split short exact sequence of $R$-modules, then for every 
$n \geq 0$ and $R$-module $N$ the sequence
\[
0 \to \Ext^n_R(M'',N) \to \Ext^n_r(M,N) \to \Ext^n_R(M',N) \to 0
\]
is also short exact and split.


\skipline
\noin
{\bf A3)} Let $R$ be an integral domain and $M$ be a
flat $R$-module.  Prove that $M$ is torsion free. Find a natural
condition on $R$ and $M$ so that the converse holds.



\skipline
\noin
{\bf A4)} Let $m,n\geq 1$ be integers. Compute
$\Tor_1^\Z(\Z/m\Z,\Z/n\Z)$.

\skipline
\noin
{\bf A5)} Let $F$, $K$, $L$ and $M$ be fields such that $F\subset K\subset M$ and
$F\subset L \subset M$, and let $KL$ denote the
subfield of $M$ generated by $K\cup L$. Write $[K:F]=m$,
$[L:F]=n$ and $[KL:F]=t$ (these cardinalities might be infinite).

\medskip
\noin
(a) Prove that $t$ is finite if and only if both $m$ and $n$ are finite.

\medskip
\noin
(b) If $t$ is finite, show that both $m$ and $n$ divide $t$, and $t\leq mn$.

\medskip
\noin
(c) If $m$ and $n$ are relatively prime, show that $t=mn$.



\skipline
\noin
{\bf A6)} If the degree of $\alpha$ over a field $F$ is odd,
prove that $F(\alpha)=F(\alpha^2)$.




\skipline
\noin
{\bf A7)} Find, with proof, the minimal polynomial over $\Q$ of $\,\root\of{2} + \root\of{3}$ and of 
$\,\root{3}\of{2} + \root\of{3}$\,.

%\ (b) $\,\root\of{2} + \root\of{3}$\, ; \ (c) $\,\root\of{2} + \root\of{3} + \root\of{5}$.



\skipline
\noin 
{\bf B1)} Let $R$ be a commutative ring.

\medskip
\noin
(a) Let  $M$, $M'$, $M''$ be $R$-modules such that there is an exact
sequence
\[
0 \to M' \to M \to M'' \to 0.
\]
If $M'$ and $M''$ are free $R$-modules, prove that $M$ is a free
$R$-module.

\medskip
\noin
(b) Let $I$ be an ideal of $R$. Show that if $I$ is free as an
$R$-module, then $I$ is a principal ideal. 

\medskip
\noin
(c) Suppose that $R$ is not a P.I.D. Show that there is an exact sequence
as in part (a) where $M$ is free but neither $M'$ nor $M''$ is free.


\skipline 
\noin 
{\bf B2)} Let $0 \to M' \to M \to M'' \to 0$, 
$0 \to A' \to P' \to M' \to 0$, 
and $0 \to A'' \to P'' \to M'' \to 0$ be exact sequences of modules over
a commutative ring. Assume that $P''$ is projective. Show that 
these sequences may be put into a commutative diagram
\[
\xymatrix{
   & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] & \\
0 \ar[r] & A' \ar[r] \ar[d] &  A \ar[r] \ar[d]
& A'' \ar[r] \ar[d] & 0 \\
0 \ar[r] & P' \ar[r] \ar[d] &  P \ar[r] \ar[d] & P'' \ar[r] \ar[d] & 0 \\
0 \ar[r] & M' \ar[r] \ar[d] &  M \ar[r] \ar[d] & M'' \ar[r] \ar[d] & 0 \\
 & 0 & 0 & 0 & }
\]
in which the rows and columns are exact and the middle row splits.







\skipline \noin {\bf B3)}  (a) Let $R$ and $S$ be commutative rings and $f:R\to S$ a 
ring homomorphism making $S$ into an $R$-module. Prove that if $S$ is
flat as an $R$-module, then $\Tor_n^R(M,N)\cong \Tor_n^S(S\otimes_RM,N)$
for all $R$-modules $M$ and $S$-modules $N$. [Hint: Show that tensoring 
an $R$-module projective resolution for $M$ with $S$ gives an $S$-module
projective resolution of $S\otimes_RM$.]

\medskip
\noin
(b) Let $D^{-1}R$ be the localization of the commutative ring $R$ with
respect to the multiplicative subset $D$ of $R$ containing $1$. Prove
that $D^{-1}R$ is flat over $R$, or equivalently, that localization 
of modules is an exact functor.

\medskip
\noin
(c) Prove that localization commutes with $\Tor$, i.e., 
\[
D^{-1}\Tor^R_n(M,N) \cong \Tor^{D^{-1}R}_n(D^{-1}M,D^{-1}N)
\]
for all $R$-modules $M$ and $N$ and all $n\geq 0$.


\medskip
\noin (d) Given any $R$-module $M$ and prime ideal $P$ of $R$, let
$R_P$ (resp.\ $M_P$) denote the localization of $R$ (resp.\ $M$) with
respect to $D=R\smallsetminus P$. Prove that an $R$-module $M$ is flat
if and only if $M_P$ is a flat for every maximal (hence also for every
prime) ideal $P$ in $R$.


\skipline

\begin{center}
{\bf \underline{Extra Credit Problem.}}
\end{center}

\skipline
\noin
{\bf C1)} Show that there does not exist a polynomial
$f(x)\in \Z[x]$ of degree greater than $1$ that is irreducible modulo $p$ for all
prime numbers $p$.






\end{document}
