# Questions tagged [algebraic-number-theory]

Algebraic number fields, Algebraic integers, Arithmetic Geometry, Elliptic Curves, Function fields, Local fields, Arithmetic groups, Automorphic forms, zeta functions, $L$-functions, Quadratic forms, Quaternion algebras, Homogenous forms, Class groups, Units, Galois theory, Group cohomology, Étale cohomology, Motives, Class field theory, Iwasawa theory, Modular curves, Shimura varieties, Jacobian varieties, Moduli spaces

2,206
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### Algorithm for finding generating sets of projective modules

Suppose $R$ is a (Dedekind) domain and $M$ is a projective module of constant rank over $R$. We know that $M$ is finitely generated over $R$. I'm wondering is there any algorithms to produce a (...

2
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0
answers

74
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### gcrd and associates of an element of the quaternion algebra over a totally real number field $K$

Let $K$ be a totally real number field of class number 1, and $Q$ the quaternion algebra over the ring of integers of $K$ with basis
$\{1,i,j,k\}$ such that $i^2 = j^2 = k^2 = -1$ and $ij = -ji, ik = ...

6
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2
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165
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### Steinitz isomorphism theorem for non-Dedekind domains

(Cross-posted from https://math.stackexchange.com/questions/4931582/steinitz-isomorphism-theorem-for-non-dedekind-domains)
Fix a Dedekind domain $R$ and fractional ideals $I, J$. It's a classical ...

0
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0
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76
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### Elliptic curve over global function field: poles of $j$-function & ramification of torsion fields [duplicate]

Let $E/ \Bbb C(t)$ be an elliptic curve over $ \Bbb C(t)$ with nonconstant $j$-invariant $j_E \in \Bbb C(t)-\Bbb C$ and $p>2$ some prime such that it is bigger than an order of a pole $v$ of $j_E$. ...

3
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1
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148
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### Arguments where the $j$-invariant has non-trivially real values — and a conjectured duality

The $j$-function (a.k.a. $j$-invariant up to a factor $1728$) is most often only considered for arguments where it yields real values.
It is well known that for $a,b,n\in\mathbb N$, $j\left(\dfrac {a+...

8
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1
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275
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### Regarding upper numbering of ramification groups

In Serre's book "Local fields" he defines the function $\phi(u)=\int_{0}^{u}\frac{dt}{( G_0:G_t)}$ and defines the upper number of ramification groups as $G^v=G_{\phi^{-1}(v)}$ and somehow ...

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### Does this subset of elliptic curves over $\mathbb{Q}$ have positive proportion?

Let $E: y^2 = x^3 + Ax + B$ be a quasi-minimal elliptic curve over $\mathbb{Q}$, i.e. $\gcd(a^3, b^2)$ is $12$th power free. Furthermore, let $\operatorname{rank}(E) = 1$ and $j(E)=\frac{1728 \times ...

2
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75
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### Distribution of smallest discriminant of number fields of degree $n$

For each positive integer $n$, define $F(n)$ to be the smallest possible absolute value of the discriminant of a number field $K/\mathbb{Q}$ of degree $n$. For example, $F(1) = 1$ and $F(2) = 3$.
By ...

8
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1
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542
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### Roadmap to Carayol-Deligne-Langlands

Having begun self-study of Fermat's Last Theorem a few years ago, I have only recently begun to understand and appreciate the theorem of Carayol-Deligne-Langlands on local-global compatibility for ...

10
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1
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795
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### Infinitely many number fields of class number 1

A classic conjecture of Gauss, which also goes by the name of "Class Number One Problem", asserts that there are infinitely many primes $p \equiv 1 \pmod{4}$ such that the quadratic field $\...

3
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0
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101
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### Describing the primes with each cyclic decomposition group in a given finite Galois extension of $\mathbb Q$

$\newcommand{\Q}{{\mathbb Q}}
$Let $f\in \Q[x]$ be a polynomial,
and let $L/\Q$ be the finite Galois extension
obtaining by adjoining to $\Q$ all roots of $f$.
Magma knows how to compute $\Gamma:={\...

2
votes

0
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244
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### Abelian extensions of number fields generated by torsion points of elliptic curve (as analogy to Lubin-Tate theory)

According to a remark from wikipedia the motivation of Lubin-Tate theory arose from the analogy to the way in which elliptic curves $E/K$ over a number field $K$ with extra endomorphisms (ie those ...

6
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0
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### Computer programs for decomposition groups?

There is quite a lot of work on computing Galois groups of splitting fields of polynomials over $\Bbb Q$. Magma is quite good at it.
In this answer to Decomposition groups for the Galois module $\mu_8$...

2
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2
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314
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### Tensor product over $\mathbb{Z}$ and p-adic integer ring $\mathbb{Z}_p$

Thanks for your reading. Suppose we have two $\mathbb{Z}_p$-modules $A,B$. Do we always have $A \otimes_{\mathbb{Z}} B \simeq A \otimes_{\mathbb{Z}_p} B$, as abelian groups or $\mathbb{Z}_p$-modules? ...

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### Is there an effective way to compute the square root of an algebraic number?

For an algebraic number $\alpha$, let $f_\alpha$ denote its minimal polynomial. We can symbolically represent an algebraic number $\alpha$ by the tuple
$$
(f_\alpha, x, y, r) \in \mathbb{Q}[x] \times ...

3
votes

1
answer

105
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### Depth of the filtration of higher ramification groups in the ramified case in Serre's modularity conjecture

I am studying Serre's paper "Sur les représentations modulaires de degré 2 de $\mathrm{Gal(\overline{\mathbb Q}/\mathbb Q)}$" and I have some questions about Serre's definition of "peu ...

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106
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### Notion of "Hodge bundle" for abelian type Shimura varieties

For a Siegel type Shimura datum $(\text{GSp}_{2g}, \mathcal{H}^{\pm})$ and level $K$, we construct the Shimura variety $S_{g,K} := \text{Sh}_K(\text{GSp}_{2g},\mathcal{H}^{\pm})$. We have a universal ...

1
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0
answers

111
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### Automorphy of the twisted representation

The Artin reciprocity says that if
$$
\chi: \operatorname{Gal}(K/\mathbb Q) \to \mathbb C
$$
is a 1-dimensional representation of a finite Galois extension $K/ \mathbb Q$, then it corresponds to a ...

7
votes

2
answers

813
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### Hilbert's Satz 90 for real simply-connected groups?

$\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Let $K/k$ be a Galois extension. Then one generalisation of Hilbert's Satz 90 states that $H^1(\Gal(K/k),\GL_n(K))=...

5
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2
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188
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### Constant term arising in general exact expression for the canonical height of a Mordell-Weil generator point on an elliptic curve $E$

First I shall begin by laying out some notation (I shall be using the conventions that are used by both DLMF and Mathematica which occasionally differ from the standard literature):
Let $\Lambda:=\...

3
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1
answer

177
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### Counting local representations for $\mathrm{GL}_2$

$\DeclareMathOperator\GL{GL}$Some context.
In number theory, it is natural to study distribution questions for the family of elliptic curves over $\mathbb{Q}$ (or any fixed number field for that ...

3
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0
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68
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### Size and usual height of an algebraic number

Let $a$ be an algebraic number of degree $n$. One defines its usual height $H(a)$ by $H(a)=\max_{0\le k\le n}|\alpha_k|$ where $P(X)=\sum_{k=0}^n\alpha_kX^k$ is the minimal polynomial of $a$ over $\...

3
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0
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106
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### The maximal $p$-abelian $p$-ramified extension of the cyclotomic $\mathbb{Z}_p$-extension

$\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\Cl{Cl}$[Reference: S. Lang, Cyclotomic Fields I and II, §2 chap 6]
Let $K$ be a number field. Suppose that $K$ contains the $p$-th roots of unity if $...

2
votes

1
answer

79
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### Cyclic extensions of a number field of full local degree in a given set $S$

Let $K$ be a number field, and let $S=S_f\cup S_{\mathbb R}\cup S_{\mathbb C}$
be a finite set of places of $K$, where $S_f$ denotes the set of finite places in $S$, $S_{\mathbb R}$ denotes the set of ...

8
votes

3
answers

718
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### About the units in $\mathbb{Z}[\frac{1+\sqrt{d}}{2}]$

Let's consider $K=\mathbb{Q}[\sqrt{d}]$ where $d$ is positive and square free. It is well known that the ring of integers is
$$
{O}_{K}=\mathbb{Z}\left[\frac{1+\sqrt{d}}{2}\right]
$$ if $d=1 \mod 4$ ...

1
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0
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117
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### On the $\mathbb{Z}_p$-rank of global Galois groups

Let $K$ be a number field and let $p$ be a prime number. Let $S_p$ denote the set of all primes of $K$ above $p$. If $S$ is finite set of primes of $K$, then we denote by $G_S$ the Galois group of ...

5
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0
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110
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### Are there only finitely many totally positive algebraic integers of bounded height and fixed absolute trace?

Let $K,t>0$ be two real constants. Consider the set $P(K,t)$ of monic polynomials with integral coefficients with all roots real, positive, and lying in the interval $(0,K)$, and with absolute ...

7
votes

1
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475
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### Field of definition of elliptic curves

Let $a,b$ be positive integers, $F=\mathbb{Q}(a^{1/3},b^{1/2})$. Let $E$ be the elliptic curve defined over $F$ by the cubic equation $$y^2=x^3+3a^{1/3}x+2b^{1/2}.$$
Then the $j$-invariant $j(E) = \...

4
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0
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196
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### Shouldn't we expect analytic (in the Berkovich sense) étale cohomology of a number field to be the cohomology of the Artin-Verdier site?

Let $K$ be a number field. Consider $X=\mathcal{M}(\mathcal O_K)$ the global Berkovich analytic space associated to $\mathcal O_K$ endowed with the norm $\|\cdot\|=\max\limits_{\sigma:K \...

6
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0
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110
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### Special value of the Artin--Mazur zeta function in arithmetic dynamics

Let $X$ be a compact manifold and $f: X\rightarrow X$ be a diffeomorphism. Assume that the $k$-fold iterate $f^k: X\rightarrow X$ has finitely many fixed points for all natural numbers $k$. The ...

13
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2
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749
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### Given an irreducible polynomial over $\mathbb{Z}$, how often is it irreducible modulo a prime?

Given a monic irreducible polynomial $f\in\mathbb{Z}[x]$, I'd like to know for how many primes p we have that $f \bmod p$ is irreducible.
In the link: How many primes stay inert in a finite (non-...

1
vote

1
answer

112
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### Grössencharakter or Galois representation associated to a CM elliptic curve in characteristic $p$

When $E$ is an elliptic curve over a number field $L$, with complex multiplication by a maximal order in an imaginary quadratic field $K$, Silverman’s Advanced Topics in the Arithmetic of Elliptic ...

5
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1
answer

244
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### Is the maximal packing density of identical circles in a circle always an algebraic number?

There is a lot of interest in the maximal density of equal circle packing in a circle. And I thought that knowing whether or not the solution is always algebraic or not would be useful.
My original ...

3
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0
answers

147
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### Proof of when 3 is a cubic residue modulo primes

I have recently been learning about cubic characters, and the machinery of Gauss and Jacobi sums used to prove the cubic reciprocity theorem, and using this, I can now determine when any prime is a ...

1
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1
answer

260
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### Integer Points on an Elliptic Curve

I am trying to find integer points $(x,y)$ on the elliptic curve
$$y^2=x^3-4x+9$$
Is there an elementary way to calculate all the solutions?
I have brute-forced solutions for $(x,y)$ under $1000$ and ...

3
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0
answers

71
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### Logarithm map for groups defined over adelic ring

I've been reading the book Eisenstein series and automorphic representations and I am struggling to understand the definition of a logarithm map $H:G(\mathbb{A})\rightarrow \mathfrak{h}(\mathbb{R})$ (...

3
votes

1
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168
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### Why locally algebraic characters of $\text{Gal}(\overline{\mathbb Q}/\mathbb Q)$ are associated to $A_0$ Grossencharacters/algebraic Hecke characters?

$\DeclareMathOperator\Res{Res}\DeclareMathOperator\Gal{Gal}$I am trying to understand lemma 3.1 of "Abelian Varieties over \mathbb Q and modular forms" of Ribet. ArXiv link
Just so everyone ...

7
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0
answers

226
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### What justifies the following isomorphism in Cassels' proof of the Cassels–Tate pairing?

In Cassels' paper Arithmetic on curves of genus 1. IV introducing the Cassels–Tate pairing the following lemma is stated.
Lemma 5.1: Let $q$ be a rational prime and $\Gamma$ the Galois group of the ...

4
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0
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91
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### Local units in a family of $S_4$-extensions

Let $a \in \mathbb{Z}$ and consider the polynomial $f(X)=X^4+aX+1$; we assume that $a$ is chosen such that $f$ is irreducible and that the discriminant $4^4-27a^4=-p$ for some prime $p$ (for example $...

7
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1
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451
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### Finiteness of the Brauer group for a one-dimensional scheme that is proper over $\mathrm{Spec}(\mathbb{Z})$

Let $X$ be a scheme with $\dim(X)=1$ that is also proper over $\mathrm{Spec}(\mathbb{Z})$. In Milne's Etale Cohomology, he states that the finiteness of the Brauer group $\mathrm{Br}(X)$ follows from ...

4
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1
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208
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### Third roots of unity and norm element

Let $K = \mathbb{Q}(\zeta_3)$ where $\zeta_3$ is a third root of unity, let $F = \mathbb{Q}(\sqrt[3]{\ell})$ where $\ell \equiv 1 \pmod{9}$ is a prime and set $L$ to be the Galois closure of $F$, i.e.,...

1
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0
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### How to prove this peculiar relationship between minimal polynomials of Ramanujan class invariants?

The Ramanujan class invariants (a.k.a. "Ramanujan-Weber class invariants")
are defined for $n>0$ by
$$G_n=2^{-1/4}e^{\pi\sqrt{n}/24}\prod_{k=0}^\infty \left(1+e^{-(2k+1)\pi\sqrt{n}}\right)...

0
votes

1
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106
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### Hansel's simple proof of the Skolem Mahler Lech theorem

In his paper 'A simple proof of the Skolem Mahler Lech theorem' Hansel gives a proof of the theorem in the case that the coefficients of the rational series belong to $\mathbb{Q}$. He claims that the ...

3
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0
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107
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### Bounding $h_3(D)$ by number of points on an elliptic curve

According to Helfgott-Venkatesh, Let $E(D)$ denote the elliptic curve $y^2 = x^3 + D$, then $h_3(Q(\sqrt D))$, which is the 3-part of the class number of the Quadratic Field with discriminant $D$, or ...

5
votes

0
answers

211
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### Applications of Langlands for GLn explicit reciprocity laws other than elliptic curves

Is there any explicit application of Langlands conjecture for $\mathrm{GL}(n)$
for $n\ge 3$, to get some reciprocity laws for higher dimensional varieties or higher genus curves?
I've never found such ...

1
vote

0
answers

35
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### Time complexity of Magma's `NormEquation` for quadratic extensions of $2$-adic fields

Note: This is similar to, but easier than, a previous question I asked here. It is a different question! I'm hoping this one might get an answer because it concerns a standard algorithm, whereas the ...

2
votes

1
answer

175
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### Artin-Schreier theorem for rings (a little different)

Motivation:
Let me recall the well-known Artin-Schreier theorem (AST) for fields in a non-formal way; if $L$ is an algebraically closed field, and $K \subset L$ a subfield not 'much smaller' than $L$, ...

16
votes

2
answers

1k
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### Is it decidable whether two real algebraic irrationals generate the same extension of the rationals?

For an algebraic number $\alpha$, let $f_\alpha$ denote its minimal polynomial. We can symbolically represent an algebraic number $\alpha$ by the tuple
$$
(f_\alpha, x, y, r) \in \mathbb{Q}[x] \times ...

4
votes

0
answers

109
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### The criterion for dimensional conjecture for universal Galois deformation rings

I’m writing to ask a question about Mazur’s dimensional conjecture in Lemma 7.5 of the paper [Galatius S, Venkatesh A. Derived Galois deformation rings. Advances in Mathematics. 2018 Mar 17;327:470-...

3
votes

1
answer

311
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### Unique factorization of ideals in a quadratic field

"Suppose $k = \mathbb{Q}(\sqrt{d})$ is a real quadratic field ($d > 1$ a square-free integer) with fundamental unit $\varepsilon$, normalized as usual so that $\varepsilon > 1$ with respect ...