Questions related to the algebraic structure of algebraic integers

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How can I choose $\frak p\unlhd\cal O$ prime so $u\in\cal O^\times$ becomes a $n$-th power (mod $\frak p$)?

$k$ is an algebraic number field, and $\cal O$ is the ring of integers, $\cal O^\times$ is the set of invertible elements of $\cal O$. Suppose $u\in\cal O^\times$ is not a $n$-th power. How can I ...
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Continuous maps from an absolute Galois group

Let $\xi$ be a continuous homomorphism from an absolute Galois group $G_{\bar{K}/K}$ (Krull topology) to a finite abelian group $M$(discrete topology), where $K$ is a number field and $\bar{K}$ is its ...
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56 views

Is solution to $\Gamma(x+1)=121$ algebraic?

If I had the following: $$x!=\Gamma(x+1)=121$$We see that $x\approx5$. But is the exact value of $x$ algebraic? For some non-whole number $x$ input that is algebraic, I think the output of the Gamma ...
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Integral elements form a ring. What can we say about polynomials of sum and product? [duplicate]

Let $B$ be a ring (commutative and with identity). It is a standard fact in Algebraic Number Theory that the sum $b_{1}+b_{2}$ and the product $b_{1}b_{2}$ of integral elements $b_{1},b_{2}\in B$ over ...
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Compare Valuations

Let $K$ be a field complete with respect to a discrete valuation $v$. Let $K_s$ its separable closure, $w$ the unique valuation extending $v$, $(\mathcal{O}_{K_s}$, $\mathfrak{M})$ respectively the ...
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fractional ideals of a number field - can they be multiplied by an integer to land in $\mathcal{O}_K$

Let $K$ be a number field (i.e a finite extension of $\mathbb{Q}$) and $\mathcal{O}_K$ its ring of integers, and $I \subseteq K$ a fractional ideal, by which I mean/define an finitely generated ...
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Neukirch ANT I.9.4: Surjective morphism $G_\mathfrak{P} \to G (\kappa(\mathfrak{P})|\kappa(\mathfrak{p}))$

I think there is a gap in the proof of this proposition and am wondering how to fix it. $L|K$ is a Galois field extension, $\mathcal{O}$ and $\mathcal{o}$ their rings of integers, $\mathfrak{P}$ is a ...
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Motivation for dual bases

I am encountering dual bases for the first time in the context of algebraic number theory, mainly in proofs regarding the existence of an integral basis for $\mathcal{O}_K$ and its ideals. I am ...
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ideal calculation and relations

Let $f$ be an integral ideal of a number field $K$ (with ring of integers $\mathcal{O}$ and let $a$ and $b$ be fractional ideals of the same. Suppose that $ab^{-1} = x\mathcal{O}$ for some $x \in K$ ...
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kernel of the artin map when dealing with S-ideles and S-divisors for function fields

Let $L/K$ be an abelian extension of function fields and let $\vartheta_{L/K}:\mathcal{D} \to \text{Gal}(L/K)$ be the Artin map from the divisors of $K$ to the galois group. What can be said about ...
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69 views

How many distinct roots of unity could have a sum of zero?

$\xi = \cos{\frac{ 2\pi}{n}}+i \sin{\frac{ 2\pi}{n}}$ , $i^2=-1, n$ is a positive integer. if $\xi^{a_1}+\xi^{a_2}+...+\xi^{a_k}=0$ , $a_1,a_2,...,a_k\in \{0,1,...,n-1\}$ and $a_1,a_2,...,a_k$ are ...
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Are there practical algorithms for computing exact eigenvalues?

Numerous software implementations exist for doing diagonalization of square matrices. However, they are iterative in nature, usually based on some fixed point equation, and returns results with ...
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1answer
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When is norm surjective mod an ideal for global fields?

Given $K/k$ Abelian, for which ideals of $\frak p\unlhd\cal O_k$ will we have $N^K_k:\cal O_K\rightarrow\cal O_k/\frak p$ surjective? $k$ is an algebraic number field. In his article "On the norm ...
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Differentiation formula in Miyake's “Modular Forms”

Miyake proves this lemma (and subsequently uses it to show the area of a hyperbolic triangle is the angle deficit): $$ (y^{-1}dz)\circ \alpha - y^{-1}dz = -2i d[log(j(\alpha,z)],$$ where $\alpha = ...
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Methods to show that an ideal isn't principal in a quadratic number field?

Suppose $a,m\in\mathbb Z_{\ge2}$. Let's consider the ring $A=\mathbb Z[(1+\alpha)/2]$, where $\alpha^2=1-4a^m$, and the ideal $I=(a,(1+\alpha)/2)$, we need to show that $I^n$ is non-principal when ...
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Minimal polynomial has coefficients in the valuation ring?

Question Let $K$ be a complete nonarchimedean valued field with valuation ring $\mathcal{O}$ and let $L/K$ be a finite extension. Let $\alpha$ be an element of $L$ and $f\in K[x]$ its minimal ...
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constructing a lattice in a number field with prescribed localizations

For a prime number $p$, we denote the localization of the ring $\mathbb{Z}$ at $p$ by $\mathbb{Z}_{(p)}$. Let $k$ be an algebraic number field and denote by $\mathfrak{o}_{k}$ its ring of integers. By ...
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$B_{\mathfrak p}$ not always a simple extension of $A_{\mathfrak p}$?

Let $B$ be the integral closure of some ring of integers $A$ in an extension of number fields, and let $\mathfrak p$ be a prime of $A$. I've seen an example where $B$ is not a simple extension of ...
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What is the correct generalization of degree of a divisor to the number field case?

When describing smooth algebraic curves over a field $k$, there are (at least) two useful notions of "class group". The first generalizes easily to general schemes: the Picard group can be defined as ...
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rank of $\mathcal{U}(\Bbb{Z}[\zeta_5])$

I have this doubt while studying Dirichlet unit theorem. By dirichlet unit theorem we know that $\mathcal{U}(\Bbb{Z}[\zeta_5])=C\times F$ where $C$ is finite cyclic and $F$ is free abelian of rank ...
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Are there non-abelian totally real extensions?

I'm aware that abelian extensions are either totally real or CM. Also, Galois extension are either totally real o totally imaginary. But I'm wondering about the converses to those statements. Are ...
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Conjugacy Class in Galois Representations

Let $G=Gal(\bar K/K)$ be the absolute Galois group of a number field $K$. Let $v$ be a finite place of $K$ and $w$ a place of $\bar K$ extending $v$. Take an $\ell$-adic representation $$ \rho: G ...
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Show that $\frac{1-\zeta_p^k}{1-\zeta_p^j}$ is invertible in $\mathbb{Z}[\zeta_p]$

Let $p$ be a prime. For $1 \leq j,k \leq p-1$, show that $$\frac{1-\zeta_p^k}{1-\zeta_p^j}$$ is invertible in the ring $\mathbb{Z}[\zeta_p]$. First, I wanted to show that ...
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1answer
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Separable extensions of complete valued fields are automatically totally ramified?

I come up with the following argument which seems to be too good to be true: Suppose that $L|K$ is finite separable extension of complete valued fields. Let $\nu, \nu'$ be the valuation on $L$ and ...
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1answer
29 views

Separability of complete value fields and residue class fields

Let $L|K$ be a finite separable extension of fields complete under some valuation and let $\lambda, \kappa$ be residue class fields of $L$ and $K$ respectively. I guess that we do not know ...
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1answer
28 views

Relationship between residue class fields between extension

Let $K$ be a field with respect to a valuation and $L$ be a finite extension. For simplicity, assume $K$ is complete. From theory, valuation on $K$ extends to $L$ and the extended valuation on $L$ is ...
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Calculation in a Group Ring

I have some problems with the verification of the third equation in Lemma 1 in this paper. First of all, one has to notice that there is at least one Error in the Definition of $a_{\kappa,\nu}$ ...
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Is a locally constant function from a profinite abelian p-group constant modulo an open normal subgroup?

Suppose $f$ is a p-adic valued function from a profinite abelian group $B$ ($Z_{p}$ or $Z_{p}^{*}$) such that for all $x$ in $B$ there is an open set $N(x)$ around $x$ such that $f$ is constant on ...
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Why does ``totally ramified'' lift as a property?

Write $D_n$ for $\mathscr{O}_{\mathbb{Q}(\zeta_n)}$. Let $\ell$ be an odd prime and $m$ an integer with $\ell \equiv 1 \text{ mod } m$. Question: Why is the following true? If the prime ideal ...
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How to prove that $V\otimes_{\mathfrak o}\mathfrak o_{\mathfrak p} \approx V\otimes_K K_\mathfrak p$

Let $\mathfrak o$ be a Dedekind domain with field of fraction $K$, $\mathfrak p$ be one of its prime ideal, $K_{\mathfrak p}$ be the completion of $K$ at $\mathfrak p$, i.e. with respect to $| ...
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A different viewpoint of Hilbert's Theorem 90

Let $L/K$ be a galois extension with galois group $G$($|G| = n$) cyclic and generated by $\sigma$. Let $\beta \in L$ have $N(\beta) = 1$. $N(.)$ is the norm function from $L$ to $K$. Hilbert's ...
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Proof of the structure theorem

I'm reading through Ian Stewart's book "Algebraic Number Theory and Fermat's Last Theorem" (3rd edition) and I'm having trouble with a bit of the proof of Theorem 1.16 (page 29). The part I don't ...
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1answer
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Characterization of cosine of rational multiples of $\pi$

Given an algebraic number $x$ such that $-1 \leq x \leq 1$ is there a characterization to figure out whether $\cos^{-1}(x)$ is a rational multiple of $\pi$ or not? One characterization would be that ...
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A consequence of “every norm is equivalent to the sup-norm” in a finite dimensional normed vector space

So, I am reading the following proposition in Neukirch's Algebraic Number Theory: Proposition. Let $K$ be a complete field with respect to the valuation $|\:\: |$ and let V be an $n$-dimensional ...
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Non-CM totally imaginary number fields

Is there a name for the totally imaginary number fields that are not CM-fields? Any important subclass of number fields with that property, or perhaps a reference where those field are studied in ...
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Is $\widehat{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{Q} \cong \mathbb{A}_{\mathbb{Q}}^f$ as topological rings?

Maybe this is rather trivial, but I could not solve this (actually, I think this is not true, however I'm not sure). Is $\widehat{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{Q} \cong ...
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Field norm of $F(\sqrt[n]{a})$

Let $F$ be a field of characteristic zero that contains a primitive $n^{th}$ root of unity. Pick $a$ such that $K=F(\sqrt[n]{a})$ is a cyclic extension of $F$ of degree $n$. Let $\sigma$ be a ...
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1answer
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Quadratic reciprocity in the case $a=-1$

I am reading the proof the for odd prime $p$, $$ \left ( \frac{-1}{p} \right)_2 = (-1)^{\frac{p-1}{2}} = \begin{cases} 1 \hspace{2mm} \text{for} \hspace{2mm} p \equiv 1 \operatorname{mod} 4 \\ -1 ...
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Properties of a Semi-modulo! operation

Let $A$ be an integer with its representation in base $p$ ($p$ may be prime number but not necessarily) described as: $$A=(a_ma_{m-1}\ldots a_1a_0)_{p}$$ We know $A\equiv (a_n\ldots a_1a_0)_{p}\pmod ...
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Basic numbers and minimisation

Given $a$ and $b$ find $c$ and $d$ such that $bc-ad$ is least and greater than zero? Also $a,b,c,d$ are integers and all lie inside a given range i.e. $[0, n]$. For example, if $n=50, a=48$ and ...
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Product of complex numbers $m+in$ with $0 < m,n \leq N$

I am trying to look for a generalization of Stirling's formula to complex numbers. In the integer case: $$ \log n! = \sum_{k = 1}^n \log k \approx \int_1^n \log x \, dx = n \log n - n$$ For the ...
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Proof of direct sum of ideal class group of Neukirch book

In books Neukirch, Algebraic Number Theory. I don't understand. 1) Why there exists $a$ such that $a\equiv c \ \mod \mathfrak p $ and $a\in ca_{\mathfrak p}^{-1}a_{\mathfrak q}$ for $\mathfrak ...
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Possible maps $(\mathbb Z[i]/\mathfrak p)^\times\to\mu_4$

Let $\mathfrak p$ be a maximal ideal of $\mathbb Z[i]$ not dividing 2. Is it true that the only maps from the cyclic group $(\mathbb Z[i]/\mathfrak p)^\times$ the the fourth roots of unity are powers ...
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square classes of quadratic extensions of 2-adic fialds

I have a question about square classes of quadratic extensions of 2-adic fields. I appreciate anybody help me to understand. Why all elements of $1+\mathfrak{p}^5$ are square in ...
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Ramification index in a number field extension

I've a basic question: Let $L/K$ be an extension of number fields, $\mathfrak p$ a prime ideal of $\mathcal O_K$ and let $\mathfrak p \mathcal O_L=\prod_i P_i^{e_i}$ where $P_i$ are primes in $L$ and ...
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Good examples of (families of) tamely ramified extensions?

I am looking for examples of families of tamely ramified extensions of $\mathbb{Q}$. Thanks.
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Is it possible to describe the places of $K$ purely in terms of the algebra of $\mathcal{O}_K$?

By "purely in terms of the algebra" of $\mathcal{O}_K$, I mean without any reference to the real numbers, topological fields, etc. Of course we can get the finite places by considering ...
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What is binary norm of quadratic fields of sum of two squares such that one of them is necessarily even like $a^2 +4b^2?$

I am trying to simplify an expression which I have reached, suppose a number can be represented in the form of $D=a^2 + 4b^2$. What is binary norm of $D$, or how else can it be represented?
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Is this polynomial irreducible over the rationals?

Prove (or disprove): Define $T_n(x)$ as the Chebyshev polynomial of the first kind with degree $n$. If $p$ is an odd prime, then $\sqrt{\frac{T_p(x)-1}{x-1}}$ is an irreducible polynomial over the ...
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Irreducible polynomials and poving the ring of intergers is a PID

My question isn't too hard I think I'm just a little stumped on how to tackle the second part. $ Let \ K=\Bbb Q(\alpha)$ where $\alpha$ is a root of $f(x)=x^3+2x+1$ 1) Show that $f(x)$ is ...