# Tagged Questions

Questions related to the algebraic structure of algebraic integers

142 views

### irreducible polynomial in $Z_p [x]$

Let $p$ be a prime integer. prove that the following are equivalent (a) $p$ is a prime in $Z[i]$ (b) $x^2 +1$ is irreducible in $Z_p [x]$ What I know is that if p is a prime in $Z[i]$, $p$ is ...
100 views

80 views

### Find all Integers ($n$) such that $n\neq 6xy\pm x\pm y$

I am interested in proving that there exist an infinite number of positive integers ($n$) which are not of the form $$n=6xy\pm x\pm y$$ for $x,y\in\Bbb Z^+$. [Note: The $\pm$ signs above are ...
43 views

### Why $\mathbb{Z}[\alpha]/23\mathbb{Z}[\alpha] \cong \mathbb{F}_{23}/(x^3-x-1)$ where $\alpha$ satisfies $\alpha^3=\alpha+1$?

This a step in my notes which I can't seem to understand clearly. Why $\mathbb{Z}[\alpha]/23\mathbb{Z}[\alpha] \cong \mathbb{F}_{23}/(x^3-x-1)$ where $\alpha$ satisfies $\alpha^3=\alpha+1$? I see ...
2k views

126 views

### The ring of integers in $\mathbb{Q}[\zeta]$ is $\mathbb{Z}[\zeta]$

I am working on a proof in the lecture of Milne "Proposition 6.2 b)" but there is a step I don't get: We have an inclusion $\mathbb{Z}\hookrightarrow \mathcal{O}_K$ that induces the following ...
170 views

164 views

### Ramification of a Galois extension

I understand that an extension of number field $L/K$ is unramified if every non-zero prime ideal of $\mathcal{O}_K$ is unramified in $L$ (where a prime ideal $\mathfrak{p}$ of $\mathcal{O}_K$ is ...
447 views

### Class group and factorizations

There is a common characterization of the class group ${\rm Cl}(R)$ as a kind of measure of how badly factorization fails to be unique. The most obvious justification for this sentiment is that the ...
164 views

### Adelic topology on the group of ideles

The topology on $\mathbb{A}^\times$ is the subspace topology with respect to $\prod_v \mathbb{Q}_v^\times$ and a basis is given by the sets $$\prod_v\Omega_v$$ with $\Omega_v\subset\mathbb{Q}_v^\times$...
383 views

### Finite abelian unramified $p$-extension of a number field

Let $K$ be a number field. How many finite abelian unramified $p$-extensions of $K$ are there and what are their Galois groups? My feeling is, that every group $\mathbb{Z} / p^n \mathbb{Z}$ can occur ...
240 views

### infinite Algebraic extensions of $\mathbb{Q}$

I need help to solve the following exercise: If $K$ is an algebraic extension of $\mathbb{Q}$ (finite or infinite), then we let $O_{K}$ denote the ring of algebraic integers in $K.$ If $\frak{P}$ is a ...
69 views

### How is this homomorphism from Algebraic Number Theory surjective?

This is on page 72 of Cassels-Fröhlich's Algebraic Number Theory. Take $S$ to be the archimedian valuations of $k$, a finite extension of $\mathbb{Q}$, and define $J_S \subset V_k$ as the subgroup ...
71 views

### Solve $f(x)\mid g^2(x)+1$ in $\mathbb Z[x]$

We know that if $p\in \mathbb P$ and $p\equiv 1\bmod 4$ then we can find $t\in\mathbb Z$ such that $p\mid t^2+1.$ For what polynomial $f(x)\in \mathbb Z[x]$, we can find $g(x)\in \mathbb Z[x]$ ...
91 views

377 views

### Are there applications of noncommutative geometry to number theory?

The marriage of algebraic geometry and number theory was celebrated in the twentieth century by the school of Grothendieck. As a consequence, number theory has been completely transformed. On the ...