Tagged Questions

0
votes
2answers
34 views

Lattice of max. rank - compact in quotient topology - bounded subset

$L\subset \mathbb{R}^{n}$ lattice of max. rank $\Leftrightarrow \mathbb{R}^{n}/L$ compact in quotient topology $\Leftrightarrow \exists$ bounded subset $B\subset\mathbb{R}^{n}$ s.t. $L+B = ...
2
votes
1answer
191 views

A non-degenerate trace implies dual basis [updated]

I found a better proof of the theorem in Serge Lang - Algebraic Number Theory but I put in bold the parts I don't understand. Hoping for any explanations of these points. The trace $Tr : L \to K$ ...
6
votes
1answer
89 views

Matrix groups generated by translation and inversion in the unit sphere

Let $\alpha$ be algebraic over $\mathbb{Q}$, and consider the subgroup $G$ of $\mathrm{SL}_2(\mathbb{C})$ generated by inversion in the unit sphere and translation by $\alpha$. That is, consider ...
2
votes
1answer
157 views

Algorithm for computing Smith normal form in an algebraic number field of class number 1

Let $K$ be an algebraic number field of class number 1. Let $\frak{O}$ be the ring of algebraic integers in $K$. Let $A$ be a nonzero $m\times n$ matrix over $\frak{O}$. Since $\frak{O}$ is a PID, $A$ ...
0
votes
0answers
42 views

Matrices with “exponential eigenvectors”?

Let $\mathcal{O}$ denote the ring of integers in some number field $k$ , and let $M= (c_{ij}) \in \mathrm{GL}_N(\mathcal{O})$ such that $$\left( \begin{matrix} 1 \\ \beta \\ \vdots \\ \beta^{N-1} ...
2
votes
0answers
201 views

Square-free discriminants and integral bases

Let $K = \mathbb Q(\alpha)$ is a number field. Suppose $\alpha \in O_K$ and let $f \in \mathbb Z[X]$ be its minimal polynomial. Show that if the discriminant of $f$ is a square-free integer, then ...
2
votes
1answer
178 views

How to see why $\Delta(x_1', \ldots , x_n ') = (\det A)^2 \Delta(x_1, \ldots ,x_n)$

Let $K$ be a number field of degree $n$, $\sigma_1 , \ldots , \sigma_n$ be the distinct embeddings of $K$ into $\mathbb C$, and define $\Delta(x_1, \ldots , x_n) = \det(\sigma_i (x_j)^2) = ...
4
votes
1answer
272 views

Why is the determinant equal to the index?

Let $A \subset B$ be integral domains and assume $B$ is a free $A$-module of rank $m$. Define the discriminant of $m$ elements $b_1,\dots,b_m\in B$ as ...