# Tagged Questions

Questions related to the algebraic structure of algebraic integers

65 views

### Probability that the discriminant of a quadratic field is divisible by a given prime number $p.$

Find the probability that the discriminant $D$ of a quadratic field $\mathbb{Q}(\sqrt{d})$ is divisible by a given prime number $p$ (beware: the result is not what you may expect.) This is an ...
216 views

66 views

106 views

### Unable to find solution for $a^2+b^2-ab$, given $a^2+b^2-ab$ is a prime number of form $3x+1$

I have a list of prime numbers which can be expressed in the form of $3x+1$. One such prime of form $3x+1$ satisfies the expression: $a^2+b^2-ab$. Now I am having list of prime numbers of form $3x+1$ ...
40 views

### Is a p-adic number field and a finite algebraic extension of it ultrametric?

An ultrametric space is a special kind of metric space in which the triangle inequality is replaced with $d(x,z)\leq\max\left\{d(x,y),d(y,z)\right\}$. Is a p-adic number field and a finite algebraic ...
442 views

### If more than one prime number satisfies a given congruence, must an infinite number of primes satisfy that congruence?

I understand that this is kind of a broad question, but if no affirmative proof is known, can anyone give a counterexample?
23 views

### Basis of neighbourhoods in a profinite group

The Krull topology in a Galois group $G$ of a Galois extension $L/K$ is defined taking $\sigma\:G(L/M)$, where $M/K$ varies through the Galois finite subextensions of $L/K$, as a fundamental system of ...
On p.239 A Course in Computational Number Theory, Cohen writes "Although the group structure on ideal classes carries over only to classes of quadratic forms via the maps $\phi_{FI}$ and $\phi_{IF}$ ...