Questions tagged [number-fields]

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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.,...
debanjana's user avatar
  • 1,201
3 votes
0 answers
50 views

On the complexity of global fields isomorphism

Let $L$ and $K$ be isomorphic global fields (i.e. either function fields of curves over a finite field, or number fields). What is the complexity of finding an isomorphism between them? Is there a ...
Reyx_0's user avatar
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2 votes
0 answers
118 views

Imaginary quadratic fields with prime class number

Let $K$ be an imaginary quadratic field, with class number equal to an odd prime, say $h_K = p$. In the proof Proposition 2.4 of this paper, Fukuda and Komatsu write, "Since $h_K = p$, there ...
matt stokes's user avatar
1 vote
1 answer
199 views

How to compute the asymptotic constant for the count of $S_3$-sextic number fields?

I am currently reading this paper counting $S_3$-sextic fields Manjul Bhargava and Melanie Matchett Wood, The density of discriminants of $S_3$-sextic number fields, Proc. Amer. Math. Soc. 136 (2008),...
gradstudent1995's user avatar
0 votes
0 answers
182 views

Distinguishing between prime factors of cubic discriminant and polynomial discriminant

Let $f(x)\in\mathbb{Q}[x]$ be an irrreducible cubic with root $\alpha$. Let $K=\mathbb{Q}(\alpha)$. There may be primes dividing $\text{disc}(f)$ that don't divide $\operatorname{disc}(K)$, so an ...
Nicolas Banks's user avatar
0 votes
0 answers
103 views

Non-isomorphic cubic fields with a given discriminant

For a cubic field $K$ with defining polynomial $P(x)=x^3 + \frac{39}{25}x^2 + \frac{22}{25}x +\frac{4}{25}$ Magma calculates the discriminant $D=-3340$. ...
Maksym Voznyy's user avatar
6 votes
0 answers
500 views

Genus of a number field

I'm reading Algebraic Number Theory by Neukirch. In chapter 3, he defines the genus of a number field as $$ g = \log \frac{ |\mu (K)| \sqrt{|d_K|}}{2^{r} (2\pi)^{s}} $$ where $|\mu(K)|$ is its ...
Leonardo Lanciano's user avatar
0 votes
0 answers
50 views

The decomposition forms of primes in $A_5$-fields

Let $K$ be a number field of degree $5$ whose Galois closure (over $\mathbb{Q}$) has the Galois group $A_5$, the alternating group of degree five. Is there any result concerning the decomposition ...
A. Maarefparvar's user avatar
1 vote
0 answers
76 views

Inflation-restrction sequence for maximal $S$-ramified extension

Let $K$ be a number field. Let $G_K$ be an absolute Galois group of $K$. Let $M$ be a $G_K$-module and $L/K$ be a finite extension. There is a inflation-restriction exact sequence, $0\to H^1(Gak(L/K), ...
Duality's user avatar
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1 vote
1 answer
96 views

Existence of a symmetric matrix satisfying certain irreducible conditions

Let $K$ be a field such that $ \mathrm{char}(K) \neq 2 $. Let $ p(x) $ be an arbitrary irreducible polynomial over $K$ of degree $n$. Using the rational canonical form, we can always construct an $ n ...
Sky's user avatar
  • 913
3 votes
1 answer
193 views

Number fields with prescriped prime decomposition

Pick your favorite prime $p$, as well as three positive integers $e,f,g$. For each such choice, does there exist at least one Galois number field $K/\mathbf{Q}$ of degree $n=efg$ in which $p$ has ...
Jeff H's user avatar
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10 votes
1 answer
398 views

Questions about ray class groups

Let $K$ be an imaginary quadratic number field (so there are no real embeddings) with ring of integers $\mathcal{O}_K$ . Let $w$ be the number of units in $K$ and $h$ be the class number of $K$. Let $\...
Joshua Stucky's user avatar
7 votes
1 answer
424 views

A cyclic Galois extension over $ \mathbb{Q}(\omega)$

It is known that $\mathbb{Q}(\sqrt{-1})$ does not live in a cyclic Galois extension $L$ of $\mathbb{Q}$ of degree $4$. For example, the image of complex conjugation in $\mathrm{Gal}(L/\mathbb{Q}) = \...
Sky's user avatar
  • 913
12 votes
0 answers
422 views

Is there a mistake in Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions"?

I am having troubles with Don Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions", available at this link, and I think there might be a mistake. In particular the ...
BernyPiffaro's user avatar
0 votes
1 answer
143 views

Existence of linear disjointness between an algebraic number field and $p$-cyclotomic field over $ \mathbb{Q}$

Suppose $ K $ be an algebraic number field and $ n $ be an even integer. Is it possible to find atleast one $ p $ such that $ p\equiv 1( \text{mod}~ n)$ and $ \mathbb{Q}(\eta_p) $ is linearly disjoint ...
Sky's user avatar
  • 913
3 votes
1 answer
778 views

irreducibility of the polynomial $ x^4 +1 $

Let $K$ be a field. We consider the polynomial $f(x) = x^4 + 1$. It is known that $f(x)$ is irreducible over $\mathbb{Q}$ but reducible over any finite field. Thus $ f(x)$ is reducible over any field $...
Sky's user avatar
  • 913
2 votes
1 answer
256 views

Number of imaginary quadratic field with its ideal class group has $\Bbb{Z}/2\Bbb{Z}$ as 2 part

Let $K=\Bbb{Q}(\sqrt{D})$($D$ is a square free negative integer) be a quadratic number field. Class number (order of ideal class group $Cl_K$ of $K$) is $1$ if only if $D=-2,-3,-7,-11,-19,-43,-67,-163$...
Duality's user avatar
  • 1,405
4 votes
1 answer
298 views

Fields in which $ -1 $ can't be written as sum of two square elements

We say a field $F$ has the property $*$ if the equation $x^2 + y^2=-1$ has no solution in $F$. For an example if $F$ is a subfield of real numbers then $F$ satisfies $*$. On the other hand if $ F $ is ...
Sky's user avatar
  • 913
0 votes
1 answer
137 views

Not containing $ 2^{i}$-th primitive roots of unity in cyclic galois extension of number field

Suppose $K$ is an algebraic number field. We want to determine whether, for all even $n$, there always exists a cyclic Galois extension $L$ of degree $n$ over $K$ such that the intermediate field $K_2$...
Sky's user avatar
  • 913
1 vote
1 answer
168 views

Norm of $2^{i}$-th primitive root

Let $ K $ be finite degree extension of $ \mathbb{Q} $ such that $ -1 $ is not a square in $ K $. Let $ L = \frac{K[x]}{\langle x^2 +1\rangle}$. Thus every element of $ L $ is of the form $ a + ib $ ...
Sky's user avatar
  • 913
6 votes
2 answers
510 views

Splitting fields of degree 4 irreducible polynomials containing a fixed quadratic extension

Let $f_1(x)\in \mathbb{Z}[x]$ be a fixed irreducible degree 4 polynomial such that its splitting field $F_1$ is an $S_4$-Galois extension over $\mathbb{Q}$ and the discriminant of $F_1$ is of the form ...
debanjana's user avatar
  • 1,201
2 votes
0 answers
117 views

$K(S,2)=\{b \in K^{\times}/{K^{\times}}^2\mid v(b)≡0 \bmod2, \forall v \notin S\}$ and Selmer group

This question is essentially related to the theory of elliptic curves (Selmer group), but this question itself is just a field theoretic one. To calculate the Selmer group of given elliptic curve, we ...
Duality's user avatar
  • 1,405
2 votes
1 answer
266 views

On Elkies' $\text{9T32}$ nonic and a shared property with j-function formulas

I. First Set Before going to Elkies' nonic, we start with something a bit simpler. There is a list of j-function formulas in this MSE post. For example, for prime levels $p = 5,7,13,$ we have, $$j=\...
Tito Piezas III's user avatar
1 vote
0 answers
152 views

The map from the decomposition group to the Galois group of the residue fields

$\DeclareMathOperator\Gal{Gal}$Let $L/K$ be a Galois field extension. Let $L$ be a field and $K$ be a number field. Let $B$ be a valuation subring of $L$ and let $A$ be the preimage of $B$ in $K$ (i.e ...
M. K.'s user avatar
  • 45
1 vote
1 answer
137 views

The map from the ring of integers to the residue field of a valuation subring is surjective

Let $L$ be a number field and let ${\mathcal{O}}_L$ be its ring of integers. Let $B$ be a valuation subring of $L$ and let $k_B$ be the residue field of $B$. Then the map from ${\mathcal{O}}_L$ to $...
M. K.'s user avatar
  • 45
1 vote
0 answers
38 views

Natural Density of Norms of ideals in a given ideal class

Some time ago, Landau proved the following formula for general number fields: $I_K(x)=U_Kx+O(x^{\delta})$, where $\delta=1-2/(1+[K:\mathbb{Q}])$, where $I_K(x)$ is the number of ideals with norm below ...
George Bentley's user avatar
3 votes
3 answers
570 views

Irreducibility of polynomials over some number fields

Recently, a problem in my research has appeared and now I need to construct some algebraic numbers with special properties (related to its degree and some other fields extensions). Now, in order to ...
Jean's user avatar
  • 515
5 votes
1 answer
246 views

Relation between $G_{\mathbb{Q}_p}$ for different primes

Let $G_{\mathbb{Q}_p}$ denote the absolute Galois group of the $p$-adic field $\mathbb{Q}_{p}$. Also, their structure as abstract groups is completely known. It is well known that this group embeds ...
kindasorta's user avatar
  • 1,661
3 votes
0 answers
122 views

Embedding number fields in fields with class number prime to $p$

Let $p$ be a fixed prime. Question: For any number field $K$, is there always a finite extension $L$ of $K$ of $p$-power order such that the class number of $L$ is prime to $p$? Moreover, for any ...
stupid boy's user avatar
4 votes
1 answer
506 views

Proving $\zeta_K\left(\frac12\right)\neq 0 \implies \zeta_K'\left(\frac12\right)\neq0?$

For an algebraic number field $K$, let $\zeta_K(s)$ be the Dedekind zeta function associated to $K$, and let $\zeta_K'(s)$ be its derivative. I believe that the following statement is true: $$\zeta_K\...
Permutator's user avatar
1 vote
0 answers
74 views

Liouville numbers with some "special" convergents

Recall that a real number $\ell$ is called a Liouville number, if there exists an infinite sequence of rational numbers $(p_n/q_n)_n$ for which $$ 0<\left|\ell-\frac{p_n}{q_n}\right|<\dfrac{1}{...
Jean's user avatar
  • 515
4 votes
1 answer
235 views

Shafarevich's conjecture on Galois groups over fields ramified at finitely many places

Let $K$ be a number field i.e. a finite extension of $\mathbb{Q}$, $\overline{K}$ a fixed separable closure of $K$, and $G_K:=\mathrm{Gal}(\overline{K}/K)$ the absolute Galois group of $K$. Let $S$ be ...
Nicolas Banks's user avatar
0 votes
0 answers
118 views

Is there a number field $K$ such that $K^\times / \mathbb{Q}^\times$ is finitely generated? [duplicate]

I think I have a simple proof that the only fields with finitely generated multiplicative groups are finite. What about if we take $K$ a number field and mod out $\mathbb{Q}^\times$ from its ...
Bma's user avatar
  • 299
0 votes
1 answer
450 views

Class number of imaginary quadratic fields

Let $n$ be a positive squarefree integer, and let $h_n$ denote the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-n})$. Then, is it true that $h_n$ is odd if and only if $n$ is a ...
user492144's user avatar
1 vote
1 answer
99 views

Size of sets associated to Gaussian integers

Given a non-zero Gaussian integer $z$, we define the set $\mathcal S(z)$ containing all solutions of $ab+cd=z$ satisfying $\min(\vert a\vert,\vert b\vert)>\max(\vert c\vert,\vert d\vert)$ with $a,b,...
Roland Bacher's user avatar
-1 votes
1 answer
173 views

Set of all primes $p$ that split in $\mathbb{Q}\left(\sqrt{-k}\right)$

Let $k$ be a squarefree positive integer. We know that a prime $p$ splits in $K=\mathbb{Q}(\sqrt{-k})$ if and only if $-k$ is a quadratic residue mod $p$. My question is: can we explicitly determine ...
user491084's user avatar
9 votes
1 answer
722 views

Is a complex algebraic set with a Zariski dense subset of algebraic points already defined over the algebraic numbers?

This is a cross-post! For the original post on SE (9 upvotes, no answer) see: https://math.stackexchange.com/questions/4475853/is-a-complex-algebraic-set-with-a-zariski-dense-subset-of-algebraic-...
Luvath's user avatar
  • 145
5 votes
2 answers
304 views

Additivity of Elliptic Curve Rank over Compositum of Fields

Assume that BSD holds for number fields. Let $E/\mathbf{Q}$ be an elliptic curve. For simplicity, let's assume it has Mordell-Weil rank zero. Let $F_1/\mathbf{Q}$ and $F_2/\mathbf{Q}$ be finite, ...
Jeff H's user avatar
  • 1,412
6 votes
1 answer
549 views

Algorithm for computing whether a cubic field is monogenic?

I am interested in existing algorithms to compute whether a given non-cyclic, non-pure cubic extension $K/\mathbb{Q}$ is monogenic or not, and if so, to give me a defining polynomial for the integral ...
edward cornfoot's user avatar
9 votes
1 answer
591 views

Square root in number field

I'm trying implement an algorithm that, for an element $b$ of a number field $\mathbb{Q}(\alpha)$, if it is a square in $\mathbb{Q}(\alpha)$ (i.e., $\exists x\in\mathbb{Q}(\alpha):x^2=b$), computes ...
Tippisum's user avatar
  • 153
6 votes
2 answers
313 views

Algebraic numbers which prescribed degree which does not belong to some fields

In my research it would be great if the following result is valid. In what follows, $\overline{\mathbb{Q}}$, $\overline{\mathbb{Q}}_n$ and $\overline{\mathbb{Q}}_{<n}$ denotes the set of algebraic ...
Jean's user avatar
  • 515
11 votes
1 answer
540 views

Sign and coefficients of fundamental unit of quadratic field

Is there any way to determine whether the fundamental unit of a quadratic field has negative or positive norm, except by actually computing the unit to all of its (many) digits? And, similarly, ...
Michael Beeson's user avatar
5 votes
2 answers
148 views

Dihedral extension unramified at primes dividing order of group?

Consider dihedral Galois extensions $L/\mathbb{Q}$ of degree $n$ (and we know they exist thanks to Shafarevich), can we show there always exists an extension $L/\mathbb{Q}$ unramified at $p$, for all $...
user404920's user avatar
1 vote
0 answers
80 views

The localization map for the Mordell-Weil group of elliptic curves over finite Galois extensions

Let $L/K$ be a finite Galois extensions of number fields and $E/K$ be an elliptic curve. Denote by $\mathcal{F}$ the localization map \begin{equation} \mathcal{F}: H^1(G,E(L)) \rightarrow \bigoplus_{v ...
A. Maarefparvar's user avatar
1 vote
0 answers
83 views

How to describe the automorphisms of $\mathbb{Q}(\theta)$ that fix $\mathbb{Q}(\sqrt{-m})$

Suppose $m > 0$ is a square free integer and $m \equiv 1 $mod $4$. Then the $K = \mathbb{Q}(\sqrt{-m})$ is a subfield of $L = \mathbb{Q}(\theta)$, where $\theta$ is a primitive $4m$-th root of ...
matt stokes's user avatar
2 votes
0 answers
103 views

Number fields with given discriminant

In the special case of number fields that are the splitting fields of irreducible polynomials of degree 5 or 6 with Symmetric Galois group (so degree 120 or 720 over Q), is there a good upper bound on ...
Joe Shipman's user avatar
3 votes
0 answers
126 views

Congruence of elements implies congruence of norms for central simple algebras

I was reading Eichler's "Allgemeine Kongruenzklasseneinteilungen [...]", Crelle 1938 (one of the main historical references for strong approximation theorems), and I cannot understand one of ...
Radu T's user avatar
  • 767
4 votes
1 answer
250 views

Order of $37$-Sylow subgroup of ideal class group of $K_{37} = \Bbb Q(\mu_{37^{n}})$ is known to be $37^n$

I want to examine nontrivial examples of what we call Iwasawa class formula, $c(n)=\mu p^n + \lambda n + \nu$, where $\lambda, \mu \in \mathbf N$ and $\nu \in \mathbf Z$ are parameters depending only ...
Duality's user avatar
  • 1,405
4 votes
1 answer
410 views

Common Galois extension over $\mathbb Q $

Suppose $L'$ is a fixed cyclic galois extension over $\mathbb {Q} $ of degree $4$.Now we know that there exists also a degree $k$ extension $L$ over $L'$ but the extension $(L/\mathbb{Q})$ may not ...
Sky's user avatar
  • 913
3 votes
0 answers
181 views

Decomposition of primes in cyclotomic extensions and their ramifications

Let $p$ be a prime. Suppose $L$ is a degree $p$ Galois extension over a number field $K$. Suppose $p$ splits both in $K$ and $L$. So there will be $[K:\mathbb{Q}]$ primes of $K$ over $p$. Call them $...
user100603's user avatar

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