The number-fields tag has no wiki summary.

**17**

votes

**0**answers

790 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

**0**answers

451 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 ...

**4**

votes

**0**answers

147 views

### Degree of Kummer extensions of number fields

Let $K$ be a number field and $a\in K^*$ of infinite order in $K^*$. How do I show that
$$[K(\sqrt[n]{a},\zeta_n):K]\geq C\cdot n\cdot\varphi(n)$$
holds for all positive integers $n$, with a positive ...

**3**

votes

**0**answers

137 views

### Irreducibility of the trinomial over Q

I'm trying to find an algebraic proof of irreducibility of the polynomial $x^n-x-1$ over rational numbers (or integers, which the same). I've read the Selmer's paper "On the irreducibility of certain ...

**2**

votes

**0**answers

106 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, ...

**1**

vote

**0**answers

125 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 ...

**1**

vote

**0**answers

131 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 ...

**0**

votes

**0**answers

18 views

### sufficiency of the relative discriminant to be a square of an ideal for an unramified quadratic extension

Is it sufficient for a quadratic extension of a cubic number field to have a relative discriminant as a square of an ideal for being unramified extension (excluding primes dividing 2 for the sake of ...

**0**

votes

**0**answers

41 views

### existence of elements with specific norms in pure cubic fields

Is there any specific way to find an element with a given norm in pure cubic field? say for an example an element of norm 5 in pure (monogenic) cubic field of 11.
it is easy to check that 5 as an ...

**0**

votes

**0**answers

70 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

**0**answers

165 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?