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.,...
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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 ...
• 139
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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 ...
• 697
1 vote
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),...
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 ...
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 $D=-3340$. ...
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 ...
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 ...
1 vote
76 views

• 913
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 ...
• 1,412
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• 913
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 ...
• 149
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 ...
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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 ...
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-...
• 145
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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, ...
• 1,412
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 ...
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 ...
• 153
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 ...
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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, ...
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