# Tagged Questions

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### $e=1$ in Theorem 30 from Marcus book “number fields”

Theorem 30 in Marcus book states that, if $p\in\mathbb Z$ is an odd prime and $q$ is a prime $\neq p$, then, fixing $d$ as a divisor of $p-1$ we have that $q$ is a $d$-th power $\operatorname{mod}q$ ...
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### Showing the center of an endomorphism ring is a direct summand

I am reading A. FrÃ¶hlich's Formal Groups, and I am working on the proof that if $F$ is a formal group defined over a separably closed field $k$ of characteristic $p$, then the endomorphism ring $E$ of ...
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### An extension of an algebraic number field which makes an integral ideal $I$, a principal ideal

I want to show that, given an ideal $I \subseteq \mathcal O_K$ (where $K/\mathbb Q$ is an algebraic number field), there is a finite extension $K'/K$ such that, $I\mathcal O_{K'}$ becomes a principal ...
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### Localization of lattices over Dedekind domains

I'm trying to understand the proof of lemma 4.12 on modules over Dedekind domains from Frohlich and Taylor's book 'Algebraic Number Theory' page 94. I have a Dedekind domain $\mathcal o$, a non-zero ...
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### Is $\mathbb{Z}[\sqrt{2} + \sqrt{3}]$ closed under multiplication?

Bonus question: if it's not, is it a subdomain of some ring of algebraic integers? This is just something I was thinking about a few weeks ago. I forgot about the concept of algebraic degrees, which ...
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### Expressing $\mathcal O_L$ as a certain free module of rank 1

I have a finite Galois extension of number fields $L/K$ with group $G$ and respective rings of integers $\mathcal O_L$ and $\mathcal O_K$. If $\Gamma$ is an $\mathcal O_K$-order in $K[G]$ and ...
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### A finite Galois extension $L/K$ of number fields.

I have a finite Galois extension $L/K$ of number fields with group $G$. Let the respective rings of integers be $\mathcal O_L$ and $\mathcal O_K.$ Suppose that $\Gamma$ is an $\mathcal O_K$-order in ...
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### A certain $\mathcal O_K$-lattice where $K$ is a number field

I have a finite Galois extension of number fields $L/K$ with group $G$. Let $\mathcal O_L$ and $\mathcal O_K$ be the respective rings of algebraic integers. I want to show that $\mathcal O_L$ is ...
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### Rings of algebraic integers

A basic question on algebraic numbers. If $L/K$ is a finite extension of number fields with respective rings of integers $\mathcal O_L$ and $\mathcal O_K$ then is it true that $\mathcal O_L$ is ...
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### The prime elements of the ring $\mathbb{Z} \! \left[ \dfrac{i \sqrt{3} - 1}{2} \right]$.

I have to study the prime elements of the ring $\mathbb{Z} \! \left[ \dfrac{i \sqrt{3} - 1}{2} \right]$. For the moment, I cannot find the general form of such elements. Can you help me? Thanks! :) ...
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### Show that it doesn't exist any of natural number $n = 4m + 3$ that $n= x^2+y^2$ for any natural x and y [duplicate]

Show that it doesn't exist any of natural number $n = 4m + 3$ that $n= x^2+y^2$ for any natural x and y Show that every prime number in form $p=4m+1$ could be showed as $p = x^2+y^2$ (x and y ...
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### ideal calculations: $2\mathcal{O}_K=\mathfrak{B}^4$ in the ring of integers of $K=\mathbb{Q}(i,\sqrt{2m})$

Let $K=\mathbb{Q}(i,\sqrt{2m})$ where $m \in \mathbb{Z}$ is odd and squarefree. Let $\alpha = (1+i)\sqrt{2m}/2$. Then $\alpha^2=im$, such that $\alpha$ is part of the ring of integers $\mathcal{O}_K$. ...
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### Question on extensions of discrete valuation fields

Let $F$ be a discrete valuation field. Let $L$ be a finite extension of $F$. Let $L=F(\alpha)$ where $\alpha$ belongs to ring of integers of $L$, denoted by $O_L$. Is it always true that ...
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### $\{1,\sqrt[3]{3},(\sqrt[3]{3})^2 \}$ is integral basis for $\Bbb{Q}(\sqrt{3^3})$

prove that $\{1,\sqrt[3]{3},(\sqrt[3]{3})^2 \}$ is integral basis for $\Bbb{Q}(\sqrt{3^3})$ my aim is to show that $\Bbb{Q}_K=\Bbb{Z}+\sqrt[3]{3}\Bbb{Z}+(\sqrt[3]{3})^2\Bbb{Z}$ $\supseteq$ : ...
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### How can I prove an ideal is a product of two irreducible ones

I'm trying to solve this question: I have a guess that $(6+\sqrt{11})=(2,4+\sqrt{11})(2,-3\sqrt{11})$ using some formulas in this book page 48. However I couldn't verify if the multiplication of ...
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### Silly mistake in this number theory book

My question is very easily to be solved (at least I hope so) I think this book has a mistake: When I calculate I get $b_3\equiv -2 (\mod{2})$ which implies $b_3=0$, am I right? Another question, ...
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### Is this ring an integral domain?

I'm starting to study Algebraic number theory and I'm having problems with the first examples of this book. I'm trying to prove this is a quadratic domain, i.e., an integral domain: I'm sorry I ...
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### Normalisation of $\mathbb{Z}$ in $\mathbb{Q}(\sqrt{d})$ is UFD…

We know that the normalisation of $\mathbb{Z}$ in $\mathbb{Q}(\sqrt{d})$ where $d\in \mathbb{Z}$ is O=\mathbb{Z}[\beta], \text{$\beta=\sqrt{d}$ if $d\equiv2,3 \pmod 4$}; \ \frac{1+\sqrt{d}}{2} ...
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### Is the extension Galois if $\mathrm{Aut}(K)$ acts transitively on the non-ramified prime ideals?

Let $K/\mathbb Q$ be a finite extension such that $\mathrm{Aut}(K)$ acts transitively on the prime ideals that are not ramified above the same prime $p\in\mathbb N$. Is $K$ Galois? Thanks in advance. ...
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### The number of abolute value on $\mathbb{Q}(\sqrt{2})$

Let $|.|$ be the usual absolute value on $\mathbb{Q}$. The number of absolute value on $Q(\sqrt{2})$ extending |.| is 2 since $x^2-2=(x-\sqrt{2})(x+\sqrt{2})$ in $\mathbb{R}[x]$. Let ...
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### Free abelian group of fractional ideals

This question is from Ch.2 of Frohlich and Taylor's Algebraic Number Theory, page 42. Let $R$ be a Dedekind domain, $I_R$ the multiplicative group of fractional $R$-ideals. There is an isomorphism of ...
### Literature to the ring $\mathbb{Z}[\phi]$ where $\phi=\frac{1+\sqrt{5}}{2}$ is the golden ratio
I know few about algebraic number theory but recently I stumbled upon the ring $\mathbb{Z}[\phi]$ where $\phi = \frac{1+\sqrt{5}}{2}$ is the golden ratio. It seems to be a very interesting object to ...