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\title{Algebra Final}
\author{James Cummings (jcumming@andrew.cmu.edu)}





\begin{document}

\maketitle


Attempt {\em exactly three questions} from each of the sections ``Groups'', ``Rings and modules'' and ``Fields''.
You may not collaborate but may
use any printed or online source (please give a reference or link),
This final is due at 23:59:59 on Sun Dec 18.


\section*{Groups}


\begin{enumerate}

\item[G1] Prove that the group of invertible upper triangular $2 \times 2$ matrices with real entries is solvable. 
   Is it nilpotent?

\item[G2]
     Let $G$ be a finite group, let $H \lhd G$ and let $P$ be a Sylow $p$-subgroup
     of $H$.  Prove that $G = H N_G(P)$. Hint: given a group element $g$,
     what can you say about $P^g$?

\item[G3]
Let $G$ be a finite group, let $p$ be prime  and let $H \le G$ be a {\em $p$-subgroup}
 (that is a subgroup whose order is a power of $p$). 

Prove that 
\[
    [G : H]  \equiv [N_G(H) : H] \mod{p}.
\]

\item[G4]  Let the finite group $G$ act on the finite set $X$. 
 Denote the stabiliser of $x \in X$ by
   $G_x$, the orbit of $x$ by $O_x$, and the set $\{ x \in X : g \cdot x = x \}$ by
   $Fix(g)$.  


\begin{enumerate}
\item  Prove that $\sum_{g \in G} \vert Fix(g) \vert = \sum_{x \in X} \vert G_x \vert$.
\item  Prove that the number of orbits is
\[
     \frac{\sum_{g \in G} \vert Fix(g) \vert}{\vert G \vert}.
\]
\end{enumerate}

\item[G5] Show that in a group of order $48$, the intersection of two distinct Sylow $2$-subgroups has order $8$. 
 Classify all the groups of order 75.   


\item[G6]  Let $G$ be a finite group, $p$ a prime dividing the
    order of $G$ and $P$ a Sylow $p$-subgroup of $G$.
    Let $x$ and $y$ be elements of $Z(P)$  which are conjugate in $G$.
    Show that $x$ and $y$ are conjugate in $N_G(P)$.



\end{enumerate}
   

\section*{Rings and modules}

\begin{enumerate}

\item[R1]
     Let $k$ be a field
    and let $R$ be the PID $k[x]$, so that every $R$-module may be regarded as 
    a vector space over $k$; prove that if $M$ is a cyclic $R$-module then 
    either $dim_k(M)$ is finite or $M \simeq R$ as an $R$-module.



\item[R2]  Let $R$ be a ring and let $M$ be an $R$-module. Define $M[x]$ to be the set of polynomials
   with coefficients in $M$, and make $M[x]$ into an $R[x]$-module in the obvious way.
   Prove that if $M$ is a Noetherian $R$-module then $M[x]$ is a Noetherian $R[x]$-module.  



\item[R3]   Let $N$ be an $R$-module. 
        A submodule $M \le N$ is {\em maximal} iff $M < N$ and there are
        no strictly intermediate modules.
      
        Fill in the details of the following sketchy proof that if $N$ is 
        a nonzero fg  $R$-module then it has a maximal submodule. 

\begin{enumerate}

\item   If $I \neq R$ is an ideal of $R$, then $R/I$ (considered as an
  $R$-module) has a maximal  submodule.

\item   If $N$ is a nonzero cyclic  $R$-module, then $N$ has a maximal submodule.

\item   If $n_1, \ldots n_k$ is a generating set of minimal size $k$ for the nonzero fg
  $R$-module $N$, and $N' = R n_2 + \ldots R n_k$, then $N' < N$ and
  $N/N'$ is cyclic. 

\item   Every fg nonzero  $R$-module has a maximal submodule. 

\end{enumerate}

\item[R4]  Let $R$ be a ring and let $Spec(R)$ be the set of prime ideals of $R$.
       For each ring element $a$, let $O_a = \{ P \in Spec(R): a \notin P \}$.
       Say that a set $X$ of prime ideals is {\em open} if for every $P \in X$
       there exists $a$ such that  $P \in O_a \subseteq X$.

\begin{enumerate}


\item   Prove that $O_a \cap O_b = O_{a b}$,  $O_0 = \emptyset$, $O_1 =
  Spec(R)$. 


\item   Prove that the collection of open sets form a topology for $Spec(R)$,
  and describe it when $R = \mathbb{Z}$. 

\item    Prove that for any $R$  this topology  is compact. 

\end{enumerate} 

\item[R5]  Let $p$ be prime. 
 Let $R_n = {\mathbb Z}/p^n {\mathbb Z}$, and let $\pi_n: R_{n+1} \mapsto R_n$ 
  be the surjective HM which maps $a + p^{n+1} \mathbb{Z}$ to $a + p^n \mathbb{Z}$.
 Define a ring ${\mathbb Z}_p$ as follows:  the elements are infinite
 sequences  $(r_0, r_1, \ldots)$ such that $r_i \in R_i$ and $\pi_i(r_{i+1}) = r_i$ for all $i$.
 Addition are multiplication are defined coordinatewise 

 Prove that 
\begin{enumerate}

\item ${\mathbb Z}_p$ is uncountable.



\item ${\mathbb Z}_p$ is an ID which contains an isomorphic copy of $\mathbb
  Z$.


\item The sequence $(2, 2, 2, \ldots)$ has a square root in $\mathbb{Z}_7$.

\end{enumerate}

\item[R6]   Let $R$ be a ring. The {\em formal power series ring} $R[[x]]$ consists
    of infinite expressions $a_0  + a_1 x + a_2 x^2 + \ldots$ with the
    natural operations. Prove that if $R$ is Noetherian then $R[[x]]$ is Noetherian.


\end{enumerate}



\section*{Fields}

\begin{enumerate}

\item[F1]
    Let $F$ be the unique subfield of $\mathbb C$ which is a splitting
    field for $x^4 - 2$ over $\mathbb Q$.  Find $[F : {\mathbb Q}]$.
    Describe the Galois group of $F$ over $\mathbb Q$. Find all the
    fields which are intermediate between $\mathbb Q$ and $F$,and identify the 
    intermediate fields which are Gakois extensions of $\mathbb Q$.

    
  \item[F2]
    Let $p$ be an odd prime and let  $\zeta = e^{2 \pi i/p}$ , $\alpha = \zeta + \zeta^{-1}$.
 Show that
     ${\mathbb Q}(\alpha)$ is a Galois extension of $\mathbb Q$ and find its
 degree over $\mathbb Q$.  For $p =7$ find the 
    minimal polynomial  of $\alpha$ over $\mathbb Q$.


\item[F3] Let $F$ be a subfield of $\mathbb C$ with $[F : {\mathbb Q}]$ finite. Prove that $F$ contains
    only finitely many roots of unity (recall that a root of unity is a complex number
    $\zeta$ such that $\zeta^n = 1$ for some $n > 0$).
    

\item[F4]     
 An {\em ordered field} is a field $F$ together with a set $P \subseteq F$ of elements 
   such that (defining $-P = \{ -a : a \in P \}$)

\begin{enumerate}

\item   $P$ is closed under $+$ and $\times$ 
\item   $P \cap  -P = \{ 0 \}$.
\item   $P \cup  -P = F$.

\end{enumerate}

   Intuitively, you can think of $P$ as the set of
   field elements which have been designated as non-negative.

 Prove that 

\begin{enumerate}

\item   There is a unique set $P \subseteq \mathbb{R}$ such that $(\mathbb{R}, P)$ 
  is an ordered field.

\item   There is no set $P \subseteq \mathbb{C}$ such that $(\mathbb{C}, P)$ 
  is an ordered field.

\item   There are at least two sets  $P \subseteq \mathbb{Q}(\sqrt{2})$ such
  that $(\mathbb{Q}(\sqrt{2}), P)$ is an ordered field. 

\item   If $(F, P)$ is an ordered  field then   $F$ has characteristic zero
   and $-1$ is not a sum of squares in $F$.

\end{enumerate}
  

\item[F5] Let ${\mathbb Q} \subseteq F  \subseteq {\mathbb C}$ be such that
   $[F: {\mathbb Q}] < \infty$ and $F$ is a Galois extension of $\mathbb Q$.
   Prove that either $F = F \cap {\mathbb R}$ or $[F : F \cap {\mathbb R}] = 2$. 
   What if we drop the assumption that $F$ is a Galois extension of $\mathbb Q$? 

\item[F6]  Let $\alpha = \sqrt[3]{2}$ and let $\gamma = i \sqrt{\alpha -1}$. 

\begin{enumerate}

\item  Prove that $E = {\mathbb Q}(\gamma)$ is a radical extension of $\mathbb Q$.

\item  Find the least subfield $F$ of $\mathbb C$ such that $F$ contains $E$ and $F$ is a 
    Galois extension of $\mathbb Q$. 

\end{enumerate}
   

\end{enumerate}


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