Questions related to the algebraic structure of algebraic integers

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Is there a redundant assumption in Exercise 2, page 14, from Janusz “Algebraic Number Fields”?

The exercise says: Let $R$ be an integral domain with quotient field $K$ and let $M$ be an $R$-submodule of a finite dimensional $K$-vector space. Prove $M=\bigcap_{P} R_P M$, where the ...
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0answers
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Proof check: commutation of Galois automorphisms and complex conjugation in CM-fields

Let $K/\mathbb{Q}$ be a Galois CM-field with $Gal(K/\mathbb{Q})=:G$ and $J_\mathbb{C}$ be the complex conjugation. Since $K$ is a CM-field one can show, that $$J:=\phi^{-1}\circ J_\mathbb{C}\circ \phi=...
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2answers
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Polynomials generating the same $p$-adic fields

I wonder if the following fact is true: Pick $l\in \mathbb N$ a number and let $f,g\in \mathbb Z_p[x]$ be monic polynomials with coefficients in the ring of $p$-adic integers such that $f\equiv g \...
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1answer
56 views

Odd degree of extension field in Couveignes Square root method

I was reading the Couveignes method to find square root for Number Field Sieve(reference here page 4 first line). It says that for this method the degree of extension K/Q must be odd so that Norm(-x)=-...
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1answer
33 views

Understanding the notation of a paper

I am reading a paper on Algebraic Number Theory that says If $p$ divides the discriminant of polynomial $f$ $r$ times and there is the factorization into irreducibles $$f(x)\equiv g_1(x)\dots g_r(...
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61 views

A prime number $p$ is ramified in $\mathbb{Q}(\sqrt[p]{a})$.

Let $p$ be an odd prime number and $a\in \mathbb {Z}$ with $\sqrt[p]{a}\notin \mathbb{Z} $. Prove that $p$ is ramified in the number field $\mathbb{Q}(\sqrt[p]{a})$. My idea is to apply Dedekind's ...
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1answer
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Existence of units in number fields outside the rings of integers

Let $K/ \mathbb Q$ be a number field with $[K:\mathbb Q]=n$. Using that there exists a prime $p\in \mathbb Z$ which splits completely, that is $p\mathcal O_K=P_1...P_n$ for some distinct primes $P_i$ ...
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1answer
23 views

$\mathbb{Z}_p$-extensions of CM-fields

I am trying to prove some consequences of Iwasawa's Theorem for CM-fields. There is a sequence of CM-fields $$K=K_0\subseteq K_1 \subseteq \dots \subseteq K_\infty$$ so that $K_\infty/K$ is a $\mathbb{...
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0answers
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Classification of set of algebraic integers in $F$

Let $F$ be a field with $\mathbb{Q} \subseteq F \subseteq \mathbb{C}$, where $F/\mathbb{Q}$ is a finite abelian Galois extension. Then can we classify set of algebraic integers in $F$ ?
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Finding square root of polynomial in extension field (Number field sieve)

I was reading this paper, on page number 29, 2nd paragraph it is written that "take the coefficients of $ \gamma $ modulo q and applying an algorithm for taking square roots in the finite field Z[ $ \...
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Trace of Witt vectors

Let $\mathbb{F}_p$ be a finite field with $p$ elements, and $\kappa := \mathbb{F}_q$ an extension of $\mathbb{F}_p$ of degree $n$ with $q = p^n$. Then $W_\infty(\kappa)$ is a ring extension of $W_\...
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When does $\sum_{p\in\mathbb{P}} \frac{1}{|p|^2}$ diverges?

We know $\sum_{p\in\mathbb{P}} \frac{1}{|p|^2}$ diverges where $\mathbb{P}$ denotes set of all primes in $\mathbb{Z}[i]$ (because that sum is greater that $\sum_{p \equiv 3 \mod 4} \frac{1}{p}$, which ...
5
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1answer
54 views

Partitioning $\mathbb{P}^1(K)$ via the class group

Let $K\subset\mathbb{C}$ be a number field. There is a surjective map $\phi:\mathbb{P}^1(K)\to Cl(K)$ from the field to the class group, sending $[\alpha:\beta]$ to the class of the ideal $(\alpha,\...
5
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1answer
94 views

Maximizing area of a pentagon

Suppose $a,b,c,d,e$ are pairwise distinct positive integers. Consider a pentagon with sides $a,b,c,d,e$ and with angles maximizing its area (we assume that a pentagon with such sides exists). It is ...
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32 views

If $p$ is unramified in every subfield of $K$, does it mean $p$ is unramified in $K$?

I am wondering if $p$ being unramified in every subfield of $K$ means $p$ is unramified in $K$. Any hints?
3
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1answer
60 views

Norm of roots of unity conjugated by Galois automorphisms in CM-fields

Let $(K_n)_{n\geq0}$ be a sequence of CM-fields, so that $K_0\subset K_1\subset\dots$ with $[K_{n+1}:K_n]=p$ for all $n\geq0$. For $n\geq0$ let $W_n$ be the group of the roots of unity in $K_n$. Now ...
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1answer
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When is the norm of a number even?

In the ring $$\textbf{Z}[i],$$ if the norm of an element is divisible by $2$, then the element must be divisible by $$1 + i,$$ and vice versa. A similar result holds for $$\textbf{Z}[\sqrt 3]$$ and ...
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1answer
27 views

Class group embedding in coprime extension

Let $L/K$ be an extension of number fields of degree $n$. Assume that the class group of $K$ has order $h$. Prove that if $(h,n)=1$ the map $Cl(K)\rightarrow Cl(L)$, given by $I\rightarrow I\mathcal ...
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2answers
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Reference request: Binary quadratic forms

I am currently a first year grad student doing an independent study on topics in algebraic number theory and am currently looking at some of the properties of the polynomial $n^2 + n + A$, where $A \...
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1answer
51 views

Representative of the $2$-Sylow subgroup of an ideal class group

Let $C$ be the ideal class group of $\mathbb{Q}(\sqrt{-6})$. I already showed that the ideal $(2,\sqrt{-6})\in C$ is not principal in $\mathbb{Q}(\sqrt{-6})$, but it is principal in $\mathbb{Q}(\sqrt{...
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1answer
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Ideal which becomes a principal ideal in a higher field extension

I am working on the question of why the ideal $(2,\sqrt{-6})$ is not a principal ideal in $\mathbb{Q}(\sqrt{-6})$, but becomes one in $\mathbb{Q}(\sqrt{-6},\sqrt{2})$. To prove that it is not ...
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Show that $\sum_{d\mid f} \varphi(f/d) a^{|d|} \equiv 0 \pmod f$

This equation is correct when $f$ and $a$ are any integers. I want to show that this holds for $f,a\in K[x]$ where $K$ is any finite field. In the equation $\varphi(f)$ is defined as $|(K[x]/(f))^\...
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2answers
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If $a,b \in \mathbb{C}$ are transcendental over $\mathbb{Q}$ then is $a^b$ necessarily transcendental over $\mathbb{Q}$?

If $a,b \in \mathbb{C}$ are transcendental over $\mathbb{Q}$ then is $a^b$ is necessarily transcendental over $\mathbb{Q}$ ? In Wiki I found answer is no but I can't cook up an counter example. ...
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0answers
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Finding units in quadratic integer rings

I want to find the units in $\mathbb{Z}[\alpha]$, where $\alpha=\frac{1+\sqrt{-11}}{2}$. One can of course use norms to find the units in quadratic integer rings of the form $\mathbb{Z}[\sqrt{D}]$ ...
2
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2answers
42 views

How to find the class number of $\mathbb{Q}(\sqrt{-17})$?

I tried to calculate the class number with help of the Minkowski bound of $M \approx 5$. So if an ideal has norm $1$, it is the ring of integers. If it has norm $2$, it is $(2, 1+\sqrt{-17})$, which ...
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0answers
23 views

Embeddings of $K_v$ in $\mathbb{C}$

Let $K$ be a number field, $v$ a nonarchimedean prime, and $K_v$ the completion of $K$ at $v$. We have the embedding $K \to K_v$, and also $K \to \mathbb{C}$. I have two related questions: Is ...
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1answer
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construct a unit on $\mathbb{Z}[\sqrt[3]{7}]$?

how might I construct a unit on $\mathbb{Z}[\sqrt[3]{7}]$? Can it be done using pigeonhole principle as with square roots and Pell equation. I had been reading about the Voronoi continued fraction or ...
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1answer
29 views

Understanding a proof in Washington's “Cyclotomic Fields”

I'm working through Washington's "Cyclotomic Fields" and having a problem with the proof of Proposition 3.8, which states: Given an abelian group G, there is an everywhere-unramified extension of ...
2
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1answer
121 views

What is number theory today? [closed]

I try to explaine my problem and I hope do not disturb or annoy; I know that number theory is very vast but essentially it is divided into two parts: analytic number theory and algebraic number ...
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2answers
68 views

Galois principle for ideals

Let $L/K$ be a finite Galois extension of number fields with Galois group $G$. Determine a necessary and sufficient condition on $L/K$ to ensure that $$\{I\in \text{Id}_L,\text{ such that }\sigma (I)...
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4answers
153 views

Solutions to a quadratic diophantine equation $x^2 + xy + y^2 = 3r^2$.

Let $k,i,r \in\Bbb Z$, $r$ constant. How to compute the number of solutions to $3(k^2+ki+i^2)=r^2$, perhaps by generating all of them?
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0answers
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How a complex root $\eta$ of $x^2 + x + A$ affects the ring $\mathbb{Z}[\eta]$

While reading a statement in P. Pollack's Not Always Buried Deep: A Second Course in Elementary Number Theory I came across a statement that seemed obvious and I am wondering if I am oversimplifying ...
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2answers
60 views

Restriction from subgroup of the Galois group of max. unr, ext. $G(\tilde{K}/\mathbb{Q}_{p})$ to $G(K/\mathbb{Q}_{p})$ is surjective?

This is a question I'm struggling with for some time. Let $K$ be a finite Galois extension of $\mathbb{Q}_{p}$ and let $\tilde{K}$ denote the maximal unramified extension of $K$. We can then ...
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0answers
34 views

Inertia of an elliptic curve with potentially good reduction

Let $E/\mathbb{Q}_p$, $p\geq5$ be an elliptic curve with additive potentially good reduction. Then there is a unique, minimal, finite and totally ramified extension $K$ such that $E/K$ has good ...
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2answers
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Finding the degree of $\sqrt[3]{2} + \sqrt[3]{3}$ over $\mathbb Q$ [duplicate]

I am practicing writing down random algebraic numbers and finding their degrees over $\mathbb Q$ and have fumbled when coming to $\sqrt[3]{2} + \sqrt[3]{3}$. Mathematica tells me the minimal ...
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2answers
68 views

Galois group of $\overline{\mathbb{F}_{p}}$ gives arithmetical information for finite fields $K/\mathbb{F}_{p}$?

Let $\mathbb{F}_{p}$ be the field with $p$ elements and $\overline{\mathbb{F}_{p}}$ be its algebraic closure. For some reason, we want to understand the structure of the Galois group of such an ...
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0answers
38 views

Image of the norm map in imaginary quadratic fields

Let $K=\mathbb{Q}(\sqrt{D})$ be an imaginary quadratic field of discriminant $D<0$. I want to know the image of the norm map $$ N^K_{\mathbb{Q}}:\mathcal{O}_K\to\mathbb{Z} $$ and the values of $N^...
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0answers
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$p$ ramify in $\mathcal{O}_K$ means repeated root modulo $p$?

I read that if $p$ ramifies in $K$ the splitting field of a polynomial $f$ on $\mathbb{Q}$, then $p\mathcal{O}_K$ has a repeated factor. How does this lead to that $f$ modulo $p$ has a repeated root?
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Explicit degree two Artin L-function

Let $K$ be a cubic field, and $\zeta_K(s)$ the Dedekind zeta function. Then from here one has the factorization $$\frac{\zeta_K(s)}{\zeta_{\bf Q}(s)}=\sum_{n=1}^\infty\frac{a(n)}{n^s}$$ where $a(n)=\...
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1answer
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Nice shapes of ideals of $\mathbb{Z}[i]$ from a (lattice) geometric point of view?

If we draw the lattice for the ideal generated by $(2+i)$ in $\mathbb{Z}[i]$, and look at what is happening modulo $(2+i)$, we see a beautiful square, although it is rotated a little bit counterclock-...
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2answers
55 views

How to construct rings with a given class number?

Hi I was learning about class number and I was wondering if it is known how to construct rings for any specific class number.
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2answers
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Does the ring of integers for the field $\mathbb{Q}(\sqrt{-1+2\sqrt{2}})$ have a power basis?

Specifically I am interested in the the ring of integers for the field $\mathbb{Q}(\sqrt{-1+2\sqrt{2}})$. Does this ring of integers have a power basis? More generally, for any Salem number $s$, ...
2
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1answer
46 views

Regulator of number fields doesn't vanish

The regulator of a number field $K$ is usually presented at the beginning of books on algebraic number theory, alongside the class number group, Dirichlet unit theorem... But the only proof for the ...
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1answer
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Upper Numbering of Ramification Groups of Absolute Galois Groups for Totally Ramified Extensions

Suppose $K'/K$ is a totally ramified extension of $p$-adic fields of degree $e.$ A paper (p.9, line 15) I am reading seems to use the following formula for the upper numbering on the absolute galois ...
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1answer
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By evaluating $\sum_{t} (1+(t/p))\zeta^t$ in two ways, prove $g=\sum_{t} \zeta^{t^2}$

I would appreciate help, please, with Exercise 6.11 in "Ireland and Rosen" (self-study). By evaluating $(1)$ $\sum_{t} (1+(t/p))\zeta^t$ in two ways, prove $(2)$ $g=\sum_{t} \zeta^{t^2}$ ...
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1answer
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Subgroups of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$

If I'm understanding the main theorem of (infinite) Galois theory correctly, applied to $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, it gives us: a) all its open subgroups are $\mathrm{Gal}(\...
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1answer
45 views

Characters defined by cyclic extensions

Let $F$ be a finite cyclic extension of degree $p$ over ${\bf Q}$. As I understand it, there is a way to associate a cyclic character to this extension. How does one do this explicitly? And how far ...
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1answer
54 views

Questions that SAGE, MAGMA can answer?

I practice theoretical mathematics and I know (almost) nothing about SAGE, MAGMA. I would like to know (in general) what type of questions can I ask SAGE to do? For example, I know that given an ...
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0answers
55 views

Non-zero ideal in algebraic integers generated by two elements

I've been doing past questions for my exams next week and would like to check an answer: Let $I$ be a non-zero ideal of the algebraic integers and let $0\neq a \in I$. Show that $\exists b \in I$ ...
3
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1answer
109 views

Value group of simple field extension: Are these value groups equal?

Suppose that a field extension $L/K$ is finite, K is a Henselian field with a exponential valuation $v$, and $w$ is an extension of $v$ to $L$. (If it is necessary, we can also assume that $L/K$ is ...