0
votes
2answers
37 views

number cubic polynomials possible

Let $p(x)$ be a cubic polynomial with integral coefficients , such that $p(a)=b$, $p(b)=c$, $p(c)=a$ for $a,b,c$ being distinct integers . find number of such possible polynomials.
4
votes
0answers
39 views

Particular determinant made of powers of algebraic numbers is nonzero?

Let $P$ be a degree-two polynomial, with roots $\alpha,\beta$. Is there a simple condition on $P$ (or on $\alpha,\beta$), equivalent to the following : $$ ...
0
votes
1answer
76 views

Interlude on Traces (and another interlude on how bad of a writer Frohlich is)

I'm trying to read through Frohlich's section of Algebraic Number Theory, but this guy really goes out of his way to make sure you don't understand anything. Frohlich is probably the guy Serre is ...
4
votes
1answer
260 views

Forcing the discriminant of an integral basis to be a Carmichael number.

I was thinking about the following lemma recently. Lemma: Let $K=\mathbb{Q}(\theta)$ for some algebraic number $\theta$ and let $n=[K:\mathbb{Q}]$. If $\{\tau_1, \,\dots\,, \tau_n\}$ consists of ...
1
vote
0answers
27 views

$M\cong N$ iff $[M:N]_R$ is a principal fractional ideal

Let $R$ be a Dedekind ring, $K$ its field of fractions, $U$ a finite vector space over $K$, and $M,N$ finitely generated $R$-modules that span $U$, i.e. contain a basis of $U$. For every $\mathfrak p ...
6
votes
0answers
141 views

Coefficients in expansion of $(\sqrt[3]{2} - 1)^m$

In trying to solve $a^3 - 2b^3 = 1$ over the integers I came across the need to answer the question: when does $(1+ \sqrt[3]{2} + \sqrt[3]{2}^2)^n$ have no $\sqrt[3]{2}^2$ term in it's expansion (in ...
0
votes
2answers
124 views

What's a Dual Basis?

If $L/K$ is a finite extension of fields with $v_1, ... , v_n$ a basis, then we have an isomorphism of $K$-modules $L/K \rightarrow Hom_K(L,K)$, where a basis for $Hom_K(L,K)$ is the "dual basis" of ...
6
votes
1answer
115 views

Eigenvalues of Multiplication in algebraic number field

Finally I have a new question which is puzzling me. I am currently reading a manuscript so there is no reference or link I can provide. I will try to put all the information needed in here. This ...
8
votes
1answer
97 views

What's special about characteristic 2?

I'm trying to get the big picture of how bilinear forms and quadratic forms relate over fields $F$ with $char(F) = 2$ and fields with $char(F) \neq 2$. What I gather so far is that if $char(F) \neq ...
0
votes
2answers
43 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
293 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
91 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
234 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
48 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
291 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
2answers
254 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
411 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 ...