Tagged Questions
3
votes
1answer
87 views
Classifying all ideals of a lattice $\mathbb{Z}[\sqrt{-d}]$
In Artin's Algebra he presents a method (that I am sure I am butchering) for classifying ideals of a given lattice $\mathbb{Z}[\sqrt{-d}]$ by taking any ideal $I$, choosing an element of minimum norm ...
1
vote
0answers
32 views
Automorphisms of a lattice and changing to a nicer $\mathbb{Z}$-base
Suppose I have an integral lattice $L$ with an arbitrary $\mathbb{Z}$-base, equipped with a positive-definite nondegenerate symmetric bilinear form $\langle\cdot,\cdot\rangle$, and an isometry $\nu$ ...
0
votes
2answers
63 views
Finding a basis for a complex lattice given a nondivisible vector in the lattice
If I am given some lattice defined as, say
$$L=\{Az_1+Bz_2\ |\ A,B \in\mathbb{Z}\}$$
and a vector $v=az_1+bz_2$ , where $\gcd(a,b)=1$, I would like to find another vector $\,w\in L\,$ such that ...
0
votes
1answer
114 views
Index of a sublattice in a lattice and a homomorphism between them
I am asked to show that if $\phi_A$ is the homomorphism from $\mathbb{Z}^k \rightarrow \mathbb{Z}^k$ given by $\phi_A(x)=xA$ then the index of $\phi(\mathbb{Z}^k)$ in $\mathbb{Z}^k$ is finite if ...
4
votes
2answers
204 views
Elementary proof that if $A$ is $m \times n$ matrix map from $\mathbb{Z}^m$ then the map is surjective iff the gcd of det of minors is 1.
I am trying to find an elementary proof that if $\phi$ is a linear map from $\mathbb{Z}^n\rightarrow \mathbb{Z}^m$ represented by $A$, an $m \times n$ matrix the map is surjective iff the gcd*strong ...
0
votes
2answers
75 views
Relating lattice bases in $\mathbb R^2$
this is a homework question but I am pretty confused on it--just don't know where to start.
We're given a lattice basis $(a, b)$ for a lattice $L$ in $\mathbb{R}^2$, and are supposed to show that ...
