Questions tagged [algebraic-number-theory]

Questions related to the algebraic structure of algebraic integers

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Sum of roots of unity with specific form

Let $m\geq 4$ be an integer. Set $\zeta:=\exp(2\pi i/m)$ and $r:=\lfloor m/2\rfloor +1$. For a positive integer $4\leq k\leq m$, we define the $S_m(k)$ as $$ S_m(k):=1+\zeta^{r}+\zeta^{2r}+\zeta^{3r}+\...
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Cubic non-residue calculation

I am currently studying Cubic residue characters from Kenneth Ireland and Michael Rosen's "A Classical Introduction to Modern Number Theory", and this is the definition given in the book: If ...
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Eisenstein integers with norm value of 5

I am trying to find out the Eisenstein integers that have norm values of 3,5,7.... I want to see if there is some pattern. For the norm value of 3, I was able to find six Eisenstein integers. For the ...
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Why is $\wedge ^2 E[p] \cong \mu_p$?

As the title says, why is $$\wedge ^2 E[p] \cong \mu_p,$$ where $E[p]$ refers to the $p$-torsion points of an Elliptic curve over a number field $K$, $\wedge$ refers to the exterior product and $\...
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Elementary proof that If $n+1$ is prime in $\mathbb{Z}$, then $1+\sqrt{-n}$ is prime in $\mathbb{Z}[\sqrt{-n}]$?

If $n+1$ is prime in $\mathbb{Z}$, then $1+\sqrt{-n}$ is prime in $\mathbb{Z}[\sqrt{-n}]$. I believe this statement is true based on the answer here: https://math.stackexchange.com/a/3610560/ However, ...
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Integral domain containing non-prime irreducibles (hence not a UFD) where all factorizations into irreducibles are unique

An integral domain is called a UFD if (1) every non-zero non-unit element factors into irreducibles, and (2) every element that factors into irreducibles does so uniquely (up to units and order). It ...
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There exists imaginary quadratic extension which trivialized 2-part of ideal class group

Let $p$ be a negative prime number such that $p \equiv 5\pmod 8$. Let $K = \mathbb{Q}(\sqrt{p})$ and denote its ideal class group by $Cl_K$. I aim to prove that $Cl_K[2] := \{a \in Cl_K \mid 2a = 0\}$ ...
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The algebraic degree of a particular sum of algebraic numbers

Let $p$ be a prime number and let $m$ be a positive integer. We know that $\beta:=p^{-1/m}$ is an $m$-degree algebraic number (in fact, its minimal polynomial is $pX^m-1$). I would like to prove that ...
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If $\vert d\vert\geq 3$, is $-1+\sqrt{d}$ an irreducible element of $\mathbb{Z}[\sqrt{d}]$?

Let $d\in\mathbb{Z}$ be an integer which is not a square (it does not have to be squarefree, though). Question. Assume that $\vert d\vert\geq 3$ to avoid special cases. Is is true that $\pi=-1+\sqrt{d}...
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Maximal abelian unramified outside S extension of exponent m of K where S is a finite set of places of K is finite

I saw a rather sleek proof of the following fact: Let $K$ be a number field. Let $L/K$ be a maximal abelian unramified outside $S$ extension of exponent $m$ of $K$ where $S$ is a finite set of places ...
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How to compute the different ideal of the cyclotomic field extension $\mathbb{Q}(\zeta_p)/\mathbb{Q}$? [closed]

Let $p$ be a prime number, $K=\mathbb{Q}(\zeta_p)$ be the cyclotomic field extension of $\mathbb{Q}$ by adding a $p$-th root of unity. There is a notation called different ideal, which is defined to ...
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Computing Asymptotic Constant for the count of $S_3$-sextic fields

I am currently reading this paper counting $S_3$-sextic fields https://www.ams.org/journals/proc/2008-136-05/S0002-9939-07-09171-X/S0002-9939-07-09171-X.pdf by Bhargava and Wood. I'm trying to verify ...
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associate a character to an abelian extension in local class field theory

Let $K_v$ a local and $F$ a finite abelian extension of it. Two questions: Could somebody explain how local class field theory associates naturally a character of $K_v^{\times}$ to $F$. And, if $r \in ...
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Does there exist a prime ideal $\mathfrak q\notin X$ with $\mathfrak q\subseteq\bigcup_{\mathfrak p\in X}\mathfrak p$?

I'm reading "Algebraic Number Theory" by Neukirch. The following paragraph is from P.70. To end this section, we now want to compare a Dedekind domain $\mathcal O$ to the ring $$\mathcal O(...
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embed roots of unity from valued field into its residue field

Let $K_v$ non-archim valued complete local field with finite residue field $\kappa= \mathcal{O}_K / \mathfrak{m}_v$ of characteristic $p$. Assume $K_v$ contains $d$th roots of unity $U_d$. Is there ...
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Additive duality of character of a local fields

I am studying "the local Langlands conjecture for GL(2)" of Bushnell and Henniart. I am having a hard time getting into the mechanics. And I have some problems with one of the first ...
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Degree of the minimal polynomial of an algebraic number in genral

Is there an (efficient) algorithm to find the degree of a minimal polynomial (over $\mathbb{Q}$) of an arbitrary algebraic number $\alpha$, where $\alpha$ is expresses in terms of some (complicated) ...
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local fields are locally profinite

Let $F$ be a local field with ring of integer $\mathfrak{o}$ and maximal ideal $\mathfrak{p}$. It is clear that $$\mathfrak{o}\supseteq \mathfrak{p}\supseteq \mathfrak{p}^2\supseteq \dots $$ is a ...
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Is the cosine of a rational multiple of a full circular angle always an algebraic number? [duplicate]

The answer is yes. Let us state a generalized version of the formula for the cosine, which applies to all units of angular measurement, with $H$ being the measure of a half-circle, $i^2=-1$, $e$ the ...
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Computing discriminant of an elliptic curve

Let $E$ be the elliptic curve over $\mathbb{Q}$ defined by $y^2+y=x^3-x$. Show that the discriminant $\Delta=37$. Attempt: For an elliptic curve of the form $y^2=x^3+Ax+B$, the discriminant is $4A^3+...
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Unique unramified ideal implies that the ramification index is equal to the degree of field extension in a galois extension

Given a Galois extension $K \supseteq \mathbb{Q} $, prove that if there is only one unramified prime number $p$ over $K$ then there is only one prime ideal $\mathfrak{p} \subseteq O_K$ containing $p$ ...
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Factorization of ideals in number fields of the form $K=\mathbb{Q}(\alpha,\beta)$

Given a number field $K=\mathbb{Q}(\alpha)$, I am able to factorize ideals of the form $p\mathcal{O}_K$ where $p$ is some unramified prime, via the Dedekind Kummer theorem. Now I have two questions: ...
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"Explicit" showing of non-flatness for a fractional ideal

I am not very familiar with flatness so I was trying to get a feel of what precisely is the crux of the notion, using objects I am used to. Sadly, most of the counter examples I could find where for ...
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On congruence subgroups of $GL_2^+$

I am reading the book Holomorphic Hilbert Modular Forms by Paul Garrett. The author considers congruence subgroups of general linear groups of positive determinant. More precisely, let $F$ be a ...
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If $p$ and $q$ are real number and $\lfloor np \rfloor \mid \lfloor nq \rfloor$ for all positive integers $n$ than $p$ and $q$ are integers

Suppose that $p$ and $q$ are two distinct positive real numbers with the property that $\lfloor np \rfloor$ divides $\lfloor nq \rfloor$ for all positive integers $n$, where $\lfloor x \rfloor$ ...
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class number of quadratic field with discriminant $17$ [duplicate]

I'm trying to prove that class number of the quadratic field with discriminant $17$ is $1$. Let $K =Q[\sqrt{17}]$ be the quadratic field and $\mathfrak{o}_{K}=\mathbb{Z}[\frac{1+\sqrt{17}}{2}]$ be the ...
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Find the result of$\sqrt[3]{77-20\sqrt{13}}+\sqrt[3]{77+20\sqrt{13}}$ [duplicate]

How to find the result of$$\sqrt[3]{77-20\sqrt{13}}+\sqrt[3]{77+20\sqrt{13}}$$ I tried using $$\begin{align}a\pm b\sqrt{13}&=\sqrt[3]{(a\pm b\sqrt{13})^3}\\&=\sqrt[3]{(a^3+39ab^2)\pm \sqrt{13}(...
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Using algebraic geometry to calculate the ring of integers of a number field

In section 13 chapter 1 of Neukirch's Algebraic Number Theory, he shows how rings of integers can be seen as one dimensional curves, with the integral closure of a ring corresponding to the ...
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What is known about the equation $x^2+ay^2=b^2$, where $a$ is a fixed square free positive integer and $b$ is a fixed positive integer. [closed]

$(b,0)$ and $(-b,0)$ are two trivial solutions. What do we know about the nontrivial solutions of the equation $x^2+ay^2=b^2$.
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Problems in $p$-adic numbers accessible to an early undergraduate

I am going for a research internship this summer in $p$-adic numbers. I am currently taking a number theory course, and have the essentials of a first year mathematics student, like multivariable ...
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Doubt in Proposition 2.2 chapter 3Neukirch algebraic number theory

Prop2.2chapter3Neukirch In this Proposition 2.2 of Chapter 3 of Neukirch Algebraic Number Theory I can't understand if $\mathcal{O_\mathfrak P}$ is the completion of $\mathcal{O}$ with respect to the ...
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Show by Local Class Field Theory $\mathbf Q_p$ has unique Galois ext. iso. to $(Z/2Z)^2$ if $p > 2$, and unique Galois ext. iso. to $(Z/2Z)^3$ o/w.

Show that $\mathbf{Q}_p$ has a unique Galois extension isomorphic to $(Z/2Z)^2$ if $p > 2$, and that $\mathbf{Q}_2$ has a unique Galois extension isomorphic to $(Z/2Z)^3$ I have already completed ...
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Can a local field be local in more than one way?

A local field (to my knowledge) is a field equipped with a non-trivial absolute value that makes it a locally compact topological space. From this, one derives that the local fields are $\Bbb R,\Bbb C,...
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Relation between inertia group in character theory and commutative algebra

When studying character theory (specifically, of normal subgroups), one comes across the concept of the inertia group. If $N \unlhd G$, where $G$ is a finite group, then, $G$ acts on $\operatorname{...
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Irreducible polynomial in Dedekind domain

In a PID, we know that if we know that the coefficients of an irreducible polynomial don't have any common factor. Is the same true in a Dedekind domain ? $\bf{Updated: }$ or Are there irreducible ...
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Splits completely of a prime ideal

Suppose that $K$ is a number field and $\mathfrak p$ is a prime ideal non-zero. In general always exists a finite extension of $L$ of $K$ such that $\mathfrak p$ is ramified, for example $L=K(\sqrt f)$...
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Gamma integral in Dirichlet L-series

I am studying Dirichlet L-series in Algebraic Number Theory by Neukirch (Chap VII, section 2). In order to define the completed L-series of a character $\chi$ it started considering the gamma integral ...
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$(\varprojlim_n O_K/p^n) \otimes_{O_K} O_L \cong \varprojlim_n (O_K/p^n \otimes_{O_K} O_L)$

Let $L/K$ be an extension of global fields, and $O_K$, $O_L$ be respectively integer rings of $K$, $L$. Let $p$ be a prime ideal of $O_K$. When is the canonical homomorphism $$(\varprojlim_n O_K/p^n) \...
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Algebraic integers in cyclotomic field: confusion about index of $Z[\zeta]$ in $O_K$

I am taking a course in Algebraic Number Theory and trying to understand the proof of the following theorem: Theorem Let $\zeta$ be a primitive n-th root of unity, $K=\mathbb{Q}(\zeta)$ a cyclotomic ...
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How to prove if a curve is rational

I am confused in the approach to prove that an specific curve is rational. I know that it means that it is birationally equivalent to $P^1$ but when im working with concrete examples i get confused. ...
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Is maximal ideal of $\mathfrak{p}$ adic number field ideal extension of $\mathfrak{p}$?

Let $F_{\mathfrak{p}}$ be a completion of a number field $F$ at a non zero prime ideal $\mathfrak{p}$ of integer ring $\mathcal{O}$ and $\hat{\mathcal{O}}$ be its integer ring and $\hat{\mathfrak{p}}$ ...
user682141's user avatar
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Prove that for a imaginary quadratic field $K/\mathbb Q$, $Cl(K)/2Cl(K)= (\mathbb Z/2)^{r-1}$ where $r$ is the number of prime that ramify in $K$.

I want to prove that for a imaginary quadratic field $K/\mathbb Q$, $Cl(K)/2Cl(K)\simeq (\mathbb Z/2)^{r-1}$ where $r$ is the number of prime that ramify in $K$. This question has appeared on this ...
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Is there a useful/meaningful notion of a multi-variable L-function in number theory?

I recently encountered multi-variable generalizations of various classical zeta functions. For example, the multi-variable Riemann zeta function $$ \zeta(s_1, \ldots , s_r) := \sum_{0 < n_1 < \...
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A formula for $S$-units in quaternion algebras

Let $D=\left(\frac{a,b}{\mathbb{Q}}\right)$ be a quaternion algebra over $\mathbb{Q}$. Suppose $D$ is ramified at a finite set $S$ of places and $\infty\in S$. It is known that the $S$-units (the unit ...
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Group action of residue group on group cohomology

Let $G$ be a group and $M$ be $G$-module. Let $H^1(G,M)$ be first group cohomology. I heard $G/H$ acts on $H^1(H,M)$ naturally. What is the standard action ? I came upon an action * by $(\sigma*f)(g)=\...
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Find an abelian extension of 2-adic rationals such that $Gal(K/\mathbb Q_2)= \mathbb Z/2 \times (\mathbb Z /2^r)^2$.

Let $r>0$. I want to prove that there exists a extension $K / \mathbb Q_2$ such that $Gal(K/\mathbb Q_2)= \mathbb Z/2 \times (\mathbb Z /2^r)^2$. This is an exercise in Kedlaya's notes on class ...
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First group cohomology $H^1(G,A)$ of cyclic group of order $2$

Let $G=〈\sigma〉$ be a cyclic group of order $2$ generated by $\sigma$. Let $A$ be $G$-module. Let $H^1(G,A)$ be a first group cohomology. I want to prove $H^1(G,A)\cong \text{ker}(\sigma +1)/(\sigma -...
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Why do we need $N$ large in Marcus $3.21$?

Exercise $3.21$ of Marcus’ “Number Fields” (an excellent book) has been giving me trouble on and off for a while now. I think I’ve finally got it but I’m suspicious of my solution since it nowhere ...
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Definition of Tate-Shafarevich group up to isomorphism of elliptic curves

Let $E/K$ be an elliptic curve. Tate-Shafarevich group $Sha(E/K)$ of $E/K$ is defined as $Sha(E/K)\stackrel{\mathrm{def}}{=} \text{ker}(H^1(G_K,E) \to \prod_{v\in M_K} {H^1(G_{K_v},E)})$. When $E/K\...
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Second group cohomology of cyclic groups

Let $M$ be a Abelian group and $G$ be cyclic group of order $2$. Let $M$ be a $G$-module. Suppose $G=\langle\sigma\rangle.$ Let $M^{G} =\{m\in M\mid \sigma m =m\}$. Define the norm map $N: M\...
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