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22
votes
0answers
1k views

Orders in number fields

Let $K$ be a degree $n$ extension of ${\mathbb Q}$ with ring of integers $R$. An order in $K$ is a subring with identity of $R$ which is a ${\mathbb Z}$-module of rank $n$. Question: Let $p$ be an ...
5
votes
0answers
478 views

Elliptic curve with no points in a number field

The following is probably well-known (I'd appreciate a link): for a field $K$ that is a finite extension of the field of rational numbers, give a polynomial $f(x,y) ∈ Q[x,y]$ of the form $y^2 − x^3 ...
2
votes
0answers
153 views

Normal basis in cyclotomic number fields

Let $p$ be an odd prime integer and let $\zeta$ be a primitive $2p$-th root of unity. Does $\alpha=1+\zeta+\zeta^{-1}+\dots+\zeta^{\frac{p-1}{2}}+\zeta^{-\frac{p-1}{2}}$ generates a normal basis of ...
2
votes
0answers
132 views

Siegel Walfisz Theorem for algebraic number fields

Is there a generalization of the Siegel Walfisz to algebraic number fields? This has been done for the prime number theorem in the prime ideal theorem.
2
votes
0answers
111 views

Lowest degree polynomial with integer coefficients yielding $1/\sqrt{2^n}$

Let $x = \cos(\pi/8) = \frac{1}{2} \sqrt{2+\sqrt{2}}$ and $y = \sin(\pi/8) = \frac{1}{2} \sqrt{2-\sqrt{2}}$. What is the lowest degree polynomial $p(x,y)$ with integer coefficients such that $p(x,y) = ...
2
votes
0answers
120 views

Comparing ideal class numbers of different orders

Let $P$ be a monic irreducible integral polynomial. Let $K=\mathbf Q[X]/(P)$ be the associated number field, $\mathcal O$ be its ring of integers and $R$ be the order $\mathbf Z[X]/(P)$. (In general, ...
2
votes
0answers
163 views

bounds for the covering radius (or diameter of the Voronoi cell) of a lattice coming from a number field

Let $K$ be a number field and $O$ an order in $K$. Denote the real embeddings of $K$ by $\sigma_1,\dots,\sigma_s$ and the non-real complex embeddings of $K$ by ...
1
vote
0answers
170 views

Is it trivial to get $200$ algebraic abc triples of equal quality $1.6978…$ over isomorphic number fields?

Got $200$ algebraic abc triples over distinct though isomorphic number fields of equal quality $1.6978...$ Strongly suspect I can get as many as I like (assuming the computations are correct). Is ...
1
vote
0answers
141 views

Full $n$-torsion of elliptic curves and the cyclotomic field of order $n$

Hi, overflowers. I have a question concerning the torsion of elliptic curves over number fields. Let us consider an elliptic curve $E$ defined over ${\mathbb Q}$. From the Weil pairing one can ...
0
votes
0answers
101 views

construct totally real cubic fields

Given some $e_i=0$ or $1$ for $1\le i \le 3$. I wish to construct a totally real cubic number field $K$ so that $\prod_{i=1}^3 sgn(\sigma_i(u))^{e_i}$ is always $1$ for any $u\in O_K^\times$ where ...
0
votes
0answers
81 views

Divisor bounds of ideals in number fields

Let $K$ be an algebraic number field and let $I$ be an ideal in $O_K$ (the ring of integers). Denote by $d(I)$ the number of ideals that divide $I$. So if $I= \prod_{i=1}^k p_i^{e_i}$ is the ...
0
votes
0answers
172 views

Ring of Integers as subring with most irreducibles

Let $L$ be a number field. Is it possible to define its ring of integers $R$ by saying it's the subring with (in a fuzzy sense) the "most" irreducibles?