1
vote
3answers
61 views

Explain why the determinant of $A$ is the index of the subring?

Let $a$ be an algebraic number, whose minimal polynomial has integral coefficients. Let $K = \Bbb Q(a)$ be an algebraic number field. Let $\mathcal O_K$ be the ring of integers in this algebraic ...
1
vote
0answers
62 views

Can a ring of integers be free over a non-PID?

Let $K \subseteq L$ be an extension of number fields, and $A \subseteq B$ the corresponding rings of integers. $B$ is an $A$-module, generated by $[L : K]$ elements. If $K$ has class number one, ...
1
vote
1answer
72 views

Why is the Ideal Norm Multiplicative?

I had asked this question before and got a partial answer. Let $A$ be a Dedekind domain with quotient field $K$, $L$ a finite separable extension of $K$ of degree $n$, and $B$ the integral closure of ...
1
vote
0answers
58 views

Frobenius action on $\overline{\mathbb Q_p}$

Let $p$ be a prime number and let $F_p$ be the Frobenius automorphism of $\overline{\mathbb F_p}$. Given an explicit element $x $ of $\overline{\mathbb Q_p}$, how do I compute $F_p(x)$? Does it even ...
1
vote
1answer
111 views

Minimal polynomials and degree of field extension

I have a cyclotomic field $\mathbb{Q}(\zeta_3)$, and want to know how I can find a minimal polynomial of $\zeta_{10}$, and $\zeta_{12}$. I have determined that both the polynomials should be of ...
0
votes
0answers
25 views

ANT Frohlich Proposition 3, (v). Induced map of dual modules has the same determinant

$R$ is a Dedekind domain, $V$ is an $n$ dimensional vector space over its quotient field $K$, $B(-,-)$ is a $K$-bilinear form on $V$, and $M, N \subseteq V$ are free $R$-modules of rank $n$. Also ...
0
votes
2answers
42 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
44 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
81 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
270 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 ...
2
votes
0answers
30 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
158 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
139 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
127 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
102 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
44 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 = ...
3
votes
1answer
363 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
94 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
253 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
50 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
321 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
276 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) = ...
5
votes
1answer
471 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 ...