# Tagged Questions

Questions related to the algebraic structure of algebraic integers

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### Show that $\xi^3\equiv \pm 1 \pmod{\lambda^4}$ in $\Bbb Z [\omega]$

We have $\lambda=1-\omega$ where $\omega=e^{i 2\pi/3}$ and $\xi$ an Eisenstein integer. Given that $\xi \equiv \pm 1 \pmod{\lambda}$, how can I prove that $$\xi^3\equiv \pm 1 \pmod{\lambda^4}$$ I ...
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### An problem of ideal splitting in number field extension

If $L/K$ is Galois extension of number field, $\mathfrak{p}$ is an prime integral ideal of $K$. One would asserts that: $\mathfrak{p}\mathcal{O}_L=\mathfrak{P}_1^{e_1}\dots\mathfrak{P}_g^{e_g}$. ...
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### Monotonic roots

Consider we have a stricktly increasing positive sequence $\lambda_n$ and the following sixth order algebraic equation for every $n\in \mathbb{N}$, $$\zeta s^6-s^4+\lambda_n^2=0,$$ where $\zeta$ is a ...
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### Field extension equality using Kronecker theorem

Kronecker's theorem says that a field extension can be shown as say, F(a) represented as F[x]/minimalpoly(a). Say, Q[$\sqrt{2}$]=Q[x]/$(x^{2}-2$) And a well known example is Q[$\sqrt{2}+\sqrt{3}$]=Q[...
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### Algebraic number fields in which all rational primes are inert

Is there an algebraic number field $F\supsetneq\mathbb{Q}$ such that all rational primes are inert in $\mathcal{O}_F$?
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### Are the sets of valuations uniquely determined [closed]

Let $\mathcal{V_K}$ be the set of valuations of a number field $K$. Can it be that $\mathcal{V_L}=\mathcal{V_K}$, for the set of valuations of another number field $L$ non-isomorphic to $K$?
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### Completions of number fields at the same prime

This is probably obvious, but I don't quite see it. Archimedean completions of different number fields are always isomorphic to the same $\mathbb{R}$ or $\mathbb{C}$. Is the same true in the non-...
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### Index of norm group of a global field

For a global field $K$ (characteristic $p$ or $0$), is there anything meaningful which could be said about the value $[K^\times:N_{L/K}L^\times]$ where $L$ is a finite extension (possibly of prime ...
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### Any natural number n can be expressed as $n = 2^a \cdot b$ where $b$ is odd. Function such that $f(n) = a$

Given that any natural number $n$, can be expressed $n = 2^a \cdot b$ where $b$ is odd. Is there a function that does not include modulo or floor functions that satisfies $f(n) = a$? Thus far I have ...
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### An equality in the proof of Proposition 3 of Section 2.7 of Pierre Samuel's Algebraic Theory of Numbers

I am reading Pierre Samuel's Algebraic Theory of Numbers. I get stuck at an equality within the proof of Proposition 3 of Section 2.7. The statement of the proposition is as follows: Proposition 3. ...
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### Function field, finite extension, isomorphism implies isomorphism?

Let $A$ be a function field in $1$ variable over $\mathbb{C}$, and let $B$ be a finite extension of $A$ of degree $[B : A]$. If $B \cong \mathbb{C}(x)$ over $\mathbb{C}$, then does it necessarily ...
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### $a(x)$, $b(x) \in \mathbb{C}(x)$ and $b(x)^2 = a(x)^3 + 1$ implies $a(x)$, $b(x)$ constant?

If $a(x)$, $b(x) \in \mathbb{C}(x)$ and $b(x)^2 = a(x)^3 + 1$, then does it necessarily follow that $a(x)$ and $b(x)$ are constant? Edit. To clarify, $\mathbb{C}(x)$ is the field of rational ...
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### Criterion for the integral closure of an domain in a finite field extension being a finitely generated algebra

$A$ is an integral domain, $K=\operatorname{Frac}A$, $L/K$ finite field extension (not necessarily separable), $B$ is the integral closure of $A$ in $L$. Question: with some extra conditions on $A$, ...
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### Proof for Neukirch mysterious relationship between frobenius elements in abstract CFT

In Neukirch's ANT chapter (4) on Abstract Class Field Theory, there is a claim which I can prove, but I can't prove the "In particular" part that follows. I'm stuck in this for almost a week. I've ...
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### There are at most finitely many square-free integers $d\not\equiv 1\pmod{4}$ such that $\mathbb{Q}(\sqrt{d})$ is a Euclidean field

My book's exercise is about proving that there are at most finitely many square-free integers $d\not\equiv 1\pmod{4}$ such that $\mathbb{Q}(\sqrt{d})$ is a Euclidean field (with respect to the norm). ...
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### Norm of ideal belongs to the ideal [duplicate]

Suppose that $D$ is any number ring (i.e. $D=\mathbb Q(\alpha), \alpha \in \mathbb C$). Let $I$ be any ideal of $D$. Show that $N(I)=|D/I|$ belongs to $I$. How to start? is there a specific fact will ...
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### How to find open subgroups of finite index in $\mathbb{Q}_{3}^{\times}$?

For purposes of illustrating Local Class Field Theory, let us play with the $3$-adic numbers. I'd like to find some open subgroups of finite index in $\mathbb{Q}_{3}^{\times}$. I know about the ...
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### A line in a proof regarding nth power residues

I would appreciate help understanding this highlighted line in a proof in Ireland & Rosen (p. 45). I don't know much group theory although I know the residue classes $\pmod m$ form a ...
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### A subgroup $Z$ of $p$-adic integer such that $Z/pZ\neq C_p$.

This question may be weird in the sense that I am considering a subgroup of infinite index. Let $p$ be a prime (let's say $p\neq 2$ although I'm not sure whether this assumption is needed) and we ...
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### Using Dedekind's prime ideal factorisation theorem

I've been going over past papers for algebraic number theory and came across this question which has given me some trouble: Given a number field $K =\mathbb{Q}(\sqrt{-d})$ where $d\equiv 1 \mod 4$ ...
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### Group $U$ of $p$-adic units is inverse limit of $U/U_{n}$

In Serre's famous Course in Arithmetic, there is a somewhat unexplained claim: Let $U=\mathbb{Z}_{p}^{\times}$ be the group of $p$-adic units. For every $n\geq 1$, put $U_{n}=1+p^{n}\mathbb{Z}_{p}$...
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### What is the ring of integers in $\mathbb Q^c\otimes_K K_\mathfrak p$?

Let $K$ be a number field with ring of integers $\mathcal O_K$ and $\mathfrak p$ a prime of $K$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$. If $L$ is a number field ...
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### Completeness proofs for the solutions of Diophantine Equations

In general, what are the strategies for showing the completeness of a solution set for Diophantine Equations? For example, take the $\textit{Pell-type}$ equation $x^2 - dy^2 = a$. Say, you have a set ...
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### Significance of the Riemann hypothesis to algebraic number theory?

Of course, the truth of the Riemann hypothesis is a central question in analytic number theory. Does its truth/falsehood have important consequences in purely algebraic number theory as well? Moreover,...
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### Ramified primes in radical extension of number fields

Let $K$ be a number field, $n\ge2$ be a positive integer and $a \in K^*$. How does one show in the simplest possible way that a prime ideal $\mathfrak {p}$ of $K$ not dividing $n$ is ...
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### Adelic definition of “canonical divisor”

For a function field over a curve $F/K$, some book define the canonical divisor as the divisor of a map $\omega:\mathscr{A}_{F}\rightarrow K$ (where $\mathscr{A}_{F}$ is the pre-adele, ie. adele but ...
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### Does a place $v$ of a number field $K$ ramify in $L/K$ iff $v\mid d_L$?

Let $L/K$ be an extension of number fields and $v$ be a prime (an equivalence class of valuations) of $K$ and $d_L$ the absolute discriminant of $L$. I know that a rational prime $p$ in $\mathbb Q$ ...
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### formal derivative algebraic [closed]

Let $q$ be a prime power, $\mathbb{F}_q$ the field with $q$ elements and $f \in \mathbb{C}_\infty$ be of the form $f = \prod_{i=1}^\infty f_i$ with $f_i \in \mathbb{F}_q(X)$ (here $\mathbb{C}_\infty$ ...
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### Splitting of a prime in the compositum of two fields [duplicate]

Let $L$ and $M$ be two finite extensions of $\mathbb{Q}$ and let $LM$ denote their compositum. Suppose that $p$ is a rational prime that splits completely in $L$ and $M$. How can I show that $p$ ...
This is problem 3.26 (self-study) in "Ireland and Rosen" If $a,b,c \in \mathbb{Z}[\omega]$ and none are equal to zero, and $a^3 + b^3 +c^3 = 0$ , show at least one of $a,b,c$ is divisible by $... 1answer 41 views ### A question about the definition of$p$-adic pseudo-measure. Let$\mathfrak B$be a profinite abelian group and let$\Lambda(\mathfrak B)$be defined as the inverse limit$\varprojlim \mathbb Z_p[\mathfrak B/ \mathcal H]$where the inverse limit is taken with ... 1answer 64 views ### Proving an equivalence relation,$a^3\equiv 1 \pmod 9$, in$\mathbb{Z}[\omega]$I would appreciate help with two steps in solving this problem (self-study) from Ireland & Rosen (3.25) The problem states: Let$\lambda= 1-\omega \in \mathbb{Z}[\omega]$. And$a\equiv 1\pmod \...
I want to compute the prime factorizations of the ideals $\langle 4\sqrt{-14}\rangle$, $\langle 6\sqrt{-6} \rangle$ and $\langle 4\sqrt{-5} \rangle$ in the ring of algebraic integers of \$\mathbb{Q}(\...