Questions related to the algebraic structure of algebraic integers

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Factorization of extension is injective

Let A be a Dedekind domain with field of fractions K, and let B be the integral closure of A in a finite separable extension L of K. Now I want to show the map from Id(A) to Id(B) is injective. I know ...
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44 views

$\mathcal{O}_L$ free over $\mathcal{O}_K[G]$

Let $L/K$ be a galois extension of number fields. Suppose $G:=\text{Gal}(L/K)$ is abelian. If $\mathcal{O}_L$ is free as $\mathcal{O}_K[G]$-module, is it true that it has rank 1?
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63 views

Proof of Proposition 2.12 in Neukirch ANT

I'd like a reference or a direct proof of the following statement: Let $K|\mathbb Q$ be a finite extension and consider the ring of algebraic integers $\mathcal O_K$. Let $\mathfrak ...
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49 views

Find polynomial to use for prime ideal factorization

L.S., In an exercise for my algebraic number theory homework I came across the following problem: I would like to factor ideals $(2)$ and $(7)$ in $K = \mathbb{Q}(i, \sqrt{14})$. I managed to show ...
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85 views

Can there be a set of numbers, which have properties like those of quaternions, but of dimension 3? [duplicate]

We all know what the complex numbers are : basically $$R^2$$ with a specified product formula, which is $$(a,b).(c,d)=(ac-bd,ad+bc).$$Hamilton defined the quaternions, which are a wonderful set of ...
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21 views

Is $\widehat{K}L$ complete?

Let $K$ be a field and $\widehat{K}$ be a completion with respect to some valuation on $K$. Let $L$ be a finite separable extension of $K$. When regarded as a subfield of $\widehat{L}$, is ...
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57 views

Number field extension

Given a number field K show that there exists a number field extension L of K such that every ideal in K becomes a principal ideal in L.
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factor ideal $(3)$ in biquadratic field

L.S., I would like to factor $(3)$ in $K = \mathbb{Q}(\alpha)$ where $\alpha$ is a root of $f = X^4 + 4X^2 + 2$. I need this factoring as a part of an exercise I need to do from my course on ...
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47 views

prime ideals in extensions of dedekind domains

Let $A$ be a Dedekind domain and $B$ its integral closure on a finite extension of its field of fractions. Is it true that if $\mathfrak{I}$ is an ideal of $B$, then factorizing it as ...
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36 views

Localisation and fractional ideal

I read this in Algebraic Number Theory by A. Fröhlich & M. J. Taylor on p94: $\mathfrak o$ is a Dedekind domain with field of fraction $K$. Let $L, M$ be finitely generated torsion free ...
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26 views

Quadratic extension. Decomposition of primes

I know the following fact from basic number theory. Let $K=\Bbb{Q}(\sqrt{d})$ be a quadratic number field. Let $p$ be a prime. Then the fact that there is only one prime $\mathfrak{P}$ above $p$ in ...
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28 views

Two decomposition groups. Are they the same?

I am assuming the usual framework and notation of ramification theory. Let $G=\operatorname{Gal}(L/K)$. We define the decomposition group of a prime ideal $\mathfrak{q}$ above $\mathfrak{p}$ as ...
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38 views

$\mathfrak{a\subseteq b}$ and $\mathfrak{bc}=\lambda D$ then there is an ideal $\mathfrak{b}'$ such $\mathfrak{a=bb'}$

Let be $D$ a commutative domain, $\mathfrak{a,b,c}\subseteq D$ ideals. Show that: if $\mathfrak{a\subseteq b}$ and $\mathfrak{bc}=\lambda D$ then there is an ideal $\mathfrak{b}'$ such ...
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79 views

Is it possible to subtract a perfect square from another number to make it a perfect square?

A number $c$ is given. We need to find a number $0<k<c$ such that $c^2 - k^2$ is a perfect square. (if it is possible) $c$ and $k$ can be any positive integer. What I tried is- I iterated ...
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143 views

Minimal polynomial of product, sum, etc., of two algebraic numbers

The standard proof, apparently due to Dedekind, that algebraic numbers form a field is quick and slick; it uses the fact that $[F(\alpha) : F]$ is finite iff $\alpha$ is algebraic, and entirely avoids ...
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57 views

Sum of squares as Primes Class Field Theorem statements

We know that every prime $1\bmod 4$ can be written in an unique way as $a^2+b^2$ form where $a,b\in\Bbb N$. Is there a comprehensive list of other statements of form "every prime $d\mod r$ can be ...
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82 views

Good introductory book for Probabilistic Number Theory

I have a decent high school knowledge of Elementary Number Theory and it is also a subject I love to study. I have a good background in Real Analysis (not Complex Analysis) and Abstract Algbera. I ...
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20 views

Reference request: principalization theorem

Let $K$ be a number field, and $\mathbb{I}_K$ the group of ideles. The Hilbert class field $M$ of $K$ is the class field of the open subgroup $H = K^{\ast} \mathbb{I}_K^{S_{\infty}}$, where ...
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34 views

Question in proof from James Milne's Algebraic Number Theory

I'm having difficulty understanding a step in a proof from J.S. Milne's Algebraic Number Theory (link). Here $\zeta$ is a $p$th root of unity and $\mathfrak p = (1-\zeta^i)$ for any $1\leq i\leq p-1$ ...
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72 views

The field fixed by inertia group is the maximal unramified field

$K/F$ is an abelian galois extension of a number field. $\mathcal{O}_F$ (resp. $\mathcal{O}_K$) is the integer ring of $F$ (resp. $K$). Let $\mathfrak{P}$ be a prime of $\mathcal{O}_K$ and ...
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Number of solutions of $x_{0}x_{1} - x_{2}x_{3}$ over $\mathbb{F}_{p^{s}}$

I'm trying to solve the following exercise: Compute the zeta function of $x_{0}x_{1} - x_{2}x_{3}$ over $\mathbb{F}_{p}$. Well, for this, I need to find $N_{s}$, the number of solutions in the field ...
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moduli of the conjugates of a cyclotomic integer

I saw a theorem on the modulus of a cyclotomic integer: [Theorem] If every conjugate $\alpha '$of a cyclotomic integer $\alpha$ satisfies $|\alpha '|<\sqrt{2}$, then $\alpha$ is $0$ or a root of ...
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23 views

Does the field norm commute with field morphisms?

Consider a field extension $K$ of $\Bbb{Q}$, a finite extension $L$ of $\Bbb{Q}_p$ for some fixed prime number $p$, and a field morphism $\sigma \colon K \to L$ such that the diagram $\require{AMScd}$ ...
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63 views

$\log (1+x)$ when $x$ is $p$-adic

It's written when $x$ is $p$-adic integer then $\log (1+x) = \sum (-1)^{n-1}\frac{x^n}{n}$ converges, I don't understand what this statement mean. Can one please explain me ?
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Rank of non-zero integral ideal as a module

I am reading Pierre Samuel Algebraic Theory of Numbers. On pages 57, Let $K$ be a number field and let $n$ be its degree. $\sigma$ is the canonical imbedding of $K$ in $\mathbf R^{r_1} \times ...
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46 views

Finding $\mathbb Z[\zeta_p]^∗$, the group of units of $\mathbb Q(\zeta_p$)

Let $p$ be an odd prime and $\zeta_p$ be a primitive $p$-th root of unity. I'm trying to prove that $\mathbb Z[\zeta_p]^∗$, the group of units of $\mathbb Q(\zeta_p$) is $(\zeta_p)\mathbb Z[\zeta_p + ...
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81 views

Existence of induced map on Divisor Class Group?

Let $f: X \rightarrow Y$ be a morphism of noetherian, integral schemes, regular in codimension 1 (so we can talk about Weil divisors). I am wondering whether there is an induced map on divisor class ...
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55 views

Proving that an element generates $\mathcal{O}_K^*/ (\mathcal{O}_K^*)^3$

Let $K = \mathbb{Q}[x] / \langle x^3 + 2x^2 + 6x + 6\rangle$. The polynomial has a single real root and its discriminant is $-588$. Let $\alpha$ be the image of $x$ in $K$. Then how would I show that ...
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Find a counterexample to the following lemma if we change the statement slightly.

let K be an algebraic number field and let $O_K$ be its ring of integers. Lemma; Let $a,b$ be fractional ideals of $O_K$. If $b \subseteq a$ then there is an ideal $c$ such that $b=ac$. I need to ...
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Can someone prove or help me understand the following about Euclidean fields?

Why is it that if $\delta$ and $\delta'$ both divide $\alpha$ and $\beta$, and that every $\gamma$ which divides $\alpha$ and $\beta$ also divides $\delta$ and $\delta'$, then $\delta$ and $\delta'$ ...
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Are the Eisenstein integers the ring of integers of some algebraic number field? Can this be generalised?

The Eisenstein integers are $\mathbb Z[\omega]$ where $\omega$ is the primitive third root of unity. If $K$ is some algebraic number field, can $\mathcal O_K$ be isomorphic to $\mathbb Z[\omega]$? ...
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71 views

Ideals in the ring of gaussian integers of a given norm

What are the ideals in the ring of gaussian integers of a given norm, (say $20$) ? The ring of integers is $\mathbb Z[i]$ and it is a PID, so any ideal must be principal. If the ideal $I$ is ...
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Small integral representation as $x^2-2y^2$ in Pell's equation

Let $k$ be a "representable" positive integer, in the sense that $k=|x^2-2y^2|$ for some integers $x,y$. Does it necessarily follow that $k$ can also be represented with small parameters, i.e. ...
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70 views

Extension of a complete discrete valuation ring

My question came when I was reading the famous Tate's paper on $p$-divisible groups. At the beginning of chapter $(2.4)$ he cites this fact as obvious. If you take a complete discrete valuation ring ...
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Isomorphism of quotient rings

In a course on algebraic number theory, the lecturer says $$\mathcal{O}_K\cong \mathbb Z\left[\frac{1+\sqrt d}{2}\right] \cong\frac{\mathbb Z[x]}{\left( x^2-x-\frac{d-1}{4} \right)}.$$ This ...
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For which values of $d<0$ , is the subring of quadratic integers of $\mathbb Q[\sqrt{d}]$ is a PID?

The "integers" of quadratic field $\mathbb Q[\sqrt{d}]$ , for a squarefree integer $d$ , forms an integral domain . I know that for $d<0$ , the quadratic integers of the quadratic number fields ...
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72 views

How many positive solutions are there (Positive 3 tuples)?

I want to find how many positive solutions for the Diophantine equation $4x + 2y + 5z = 100$ I found a particular solution $(x,y,z) = (50,-50,0)$ then I found a general solution (basis) $s(-2,-1,2) + ...
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61 views

P-adic valuation for ideals

Let $A$ be a Dedekind domain and $\mathfrak{a},\mathfrak{b}$ be fractional ideals of $A$. Then we know that $\mathfrak{a}$ and $\mathfrak{b}$ can be decomposed into ...
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Existence of Conductor for Cyclotomic Extension (pg 200, Serge Lang A.N.T.)

Let $\zeta$ be a primitive $m$th root of unity, $m \not\equiv 2 \pmod 4$, and $K = \mathbb{Q}(\zeta)$. For $p$ prime and unramified i.e. $(m,p) = 1$, I know that the Artin symbol $(p, K/\mathbb{Q})$ ...
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43 views

$\mathcal{O}_K$ analogous to $\mathbb{Z}$?

The definition of $\mathcal{O}_K$ isn't very well explained or motivated in my textbook. Let $K$ be a field a field with $\mathbb{Q} \subset K \subset \mathbb{C}$. $\mathcal{O}_K$ consists of all ...
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False proof: no element has a prime norm in the ring $\mathbb Z[\zeta_p]$

I'm reading Fermat's Last Theorem by Harold M. Edwards. In the ring $\mathbb Z[\zeta_p]$, with $\zeta_p$ a primitive $p$-th root of unity ($p$ prime), we have the norm $Nf(\zeta_p)$ of an element ...
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59 views

Number of places over a number field

Let $\mathbb{K}$ a finite extension of $\mathbb{Q}$ and $\mathcal{O}_\mathbb{K}$ its ring of integers. Assume $\mathcal{O}_\mathbb{K}=\mathbb{Z}[\alpha]$, that is generated as a ring by a single ...
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When is $\mathbb{Q}[\alpha] \cap \mathbb{R}$ equal to $\mathbb{Q}$?

I'm interested in necessary and sufficient conditions for a nonreal algebraic integer $\alpha$ to satisfy the equality above. I know that if $\alpha$ has prime degree then the equality holds, but I ...
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61 views

Some questions about number fields

As a new beginner in algebraic number theory, I am confused with some properties of number fields. First comes some conventional notations. For any number field $K$, let $\mathcal{O}_K$ denote the ...
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Trouble understandingthat the special set $\mathbb{B}$ has the following properties

Let $\mathbb{B}:=\{\alpha\in\mathbb{C}|$The minimum polynomial of $\alpha$ lies in $\mathbb{Z}[x]\}$ In my notes for Algebraic number theory it's proven that the set $\mathbb{B}$ has the following ...
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Cover of $\Bbb P^1_k$ ($k$ sep. closed) unramified away from $\infty$ and tamely ramified at $\infty$

I'm reading a paper where the authors claim that for a separably closed field $k$ of characteristic $p>0$, there's no cover of $\Bbb P^1_k$ unramified away from $\infty$ and tamely ramified over ...
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165 views

Algebraic number theory exercise

I'm trying to do the following exercise from Neukirch's Algebraic Number Theory (exercise 2, $\S 1$, chapter 1): Show that, in the ring $\mathbb{Z}[i]$, the relation ...
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45 views

Specific question on imaginary quadratic field [closed]

How to solve the following question?! Let $K$ be an imaginary, quadratic field and let $L/K$ be a Galois extension. If $\tau$ is complex conjugation, show that: (a) $L/\Bbb Q$ is Galois iff ...
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Maximal ideal in local field and density of absolute value

Let $\mathbb{K}$ a non archimedean local field, and $\mathfrak{M}$ the maximal ideal of its integers ring. I have to show that $\mathfrak{M}^2\subset\mathfrak{M}$ implies that the absolute value ...
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Product involving Dirichlet characters $\prod_{i=1}^{\phi(m)}(1 - \frac{\chi_i(p)}{p^s}) =(1 - \frac{1}{p^{fs}})^{\frac{\phi(m)}{f}}$

While working with divisors in cyclotomic extensions of $\mathbb{Q},$ I came across this identity: Given a prime $p$ and an integer $m$ such that $(p,m) =1$, let $f$ be the smallest such the $p^f ...