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\begin{document}

%\subjclass{22E46, 22E47}

\title{Math 6370, HW 1}


\author{Gordan Savin} 
%\address{Gordan Savin, Department of Mathematics, University of Utah, Salt Lake City, UT 84112}
 % \email{savin@math.utah.edu}


%\begin{abstract}
  
%\end{abstract}

\maketitle

%\section{rational forms}  

Let $K$ be a finite filed extension of $\mathbb Q$ and $A$ the integral closure of $\mathbb Z$ in $K$ i.e. the ring of all algebraic integers in $K$. 
 Let $L$ be a degree $n$ extension of $K$. 
Let $B\supseteq A$ be a ring contained in the integral closure of $A$ in $L$. 
For $x_1, \ldots , x_n$ in $B$, let $D_{L/K}(x_1, \ldots , x_n)\in A$  be the determinant of the matrix $(\trace(x_i x_j))$. 
The relative discriminant $\mathscr D_{B/A}$ is the ideal in $A$ generated by all $D_{L/K}(x_1, \ldots , x_n)$. If $B$ is the integral closure of $A$ in $L$, then 
$\mathscr D_{B/A}$  is called the discriminant of $L$ over $K$. If $\mathscr D_{B/A}=A$, then the field $L$ is said to be unramified over $K$. 
 If $B$ is a free $A$-module of rank $n$, then 
$\mathscr D_{B/A}$ is the principal ideal generated by $D_{L/K}(x_1, \ldots , x_n)$, where $x_1, \ldots , x_n$ is a basis of $B$ over $A$. If $A=\mathbb Z$, then 
the discriminant is an integer, called the absolute discriminant, and denoted by $D_B$.  For example, 
if $d$ is a square-free integer, then the discriminant of $\mathbb Q(\sqrt{d})$ is $d$, if $d\equiv 1\pmod{4}$, and $4d$ if $d\equiv 2,3\pmod{4}$. 


\vskip 5pt 

\noindent  
1) Continuing with the setup as above.  Let $C\subseteq B$ be a subring of index $m$. Prove that $\mathscr D_{C/A} \subseteq \mathscr D_{B/A}$, and 
$D_C=m^2 D_B$. 

\vskip 5pt 
 
 \noindent 
2)  Let $B=\mathbb Z[\alpha]$ where $\alpha^3 \pm \alpha+1=0$ (two cases). Compute $D_B$. Prove that $B$ is the ring of (all) integers in the cubic field $\mathbb Q(\alpha)$. 
 

\vskip 5pt 

\noindent 
3) Let $B=\mathbb Z[ i, \frac{1+\sqrt{5}}{2}]$.  Its field of fractions is a bi-quadratic field extension of $\mathbb Q$. 
Find the absolute discriminant $D_B$  of this ring.  Find the relative discriminant of $\mathscr D_{B/A}$, where $A$ is  the ring of integers in each of the 
three intermediate field. Be careful,  $\mathbb Z[\sqrt{-5}]$ is not  a PID. 
 

\vskip 5pt 

\noindent 
4)   Let $L$ be a degree $n$ extension of $K$.  
Let $x_1, \ldots , x_n$   be a basis of $L$ over $K$.  
 Let  $M$ be a degree $m$ extension of $L$.  
Let $y_1, \ldots , y_m$ be a basis of $M$ over $L$.  Let $z_{ij}=x_i y_j$.  Prove that
\[ 
D_{M/K}(z_{ij})=N_{L/K}(D_{M/L}(y_i)) \cdot D_{L/K}(x_j)^{[M:L]}.
\]  

\vskip 5pt 

\noindent 
5) Let $C$ be the ring of integers in $\mathbb Q( i, \frac{1+\sqrt{5}}{2})$.  Let $A=\mathbb Z[i]$ or $\mathbb Z[ \frac{1+\sqrt{5}}{2}]$. Since $A$ is a euclidean domain, 
$C$ is free over $A$. Use the problems 1), 3) and 4) to compute $D_C$. Conclude that $C=\mathbb Z[ i, \frac{1+\sqrt{5}}{2}]$.


\end{document} 



$\mathrm G$ $\mathrm T$ $\mathrm A$ 

$\mathtt G$  $\mathtt T$  $\mathtt A$ 

$\mathcal G$ $\mathcal T$ $\mathcal A$ 

$\mathsf G$ $\mathsf T$ $\mathsf A$ 

$\mathtt G$  $\mathtt T$  $\mathtt A$ 

$\mathscr G$ $\mathscr T$ $\mathscr A$ 



\begin{thebibliography}{99}


  
\bibitem[OV]{OV} A. L. Onishchik and E. B. Vinberg (Eds.), 
{\em Lie Groups and Lie Algebras III: Structure of Lie Groups and Lie Algebras}, 
Springer-Verlag, Berlin Heidelberg, 1994.
  
\bibitem[Se]{Se} J.-P. Serre, {\em Galois Cohomology}, Springer-Verlag, Berlin Heidelberg, 1997.  



\end{thebibliography}

\end{document}


