# Tagged Questions

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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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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 : $$... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 = ... 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 ... 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 ... 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 ... 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} ...
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 ...