# Tagged Questions

Questions related to the algebraic structure of algebraic integers

49 views

### Is $(2, \sqrt{m})$ a principal ideal or not in the ring $\mathbb{Z}[\sqrt{m}]$ [closed]

Let $m$ be a negative even integer. In the ring $\mathbb{Z}[\sqrt{m}]$, is the ideal $(2,\sqrt{m})$ a principal ideal?
50 views

### Class number of $\mathbb{Q}(\sqrt{n})$ always even? [closed]

Let $n$ be a negative square-free even integer. Does it necessarily follow that the class number of $\mathbb{Q}(\sqrt{n})$ is even?
46 views

### Element of $\mathbb{F}_{p^2}$ of order $p^2$ raised to $p+1$th power is element of $\mathbb{F}_p$?

For any prime number $p$ and any element $a$ of the finite field $\mathbb{F}_{p^2}$ of order $p^2$, do we have$$a^{p+1} \in \mathbb{F}_p \subset \mathbb{F}_{p^2}?$$
62 views

### What does algebraic number look like locally?

Is there any theorem characterizing what algebraic number looks like locally (in completion)? For example, do all algebraic numbers live in some $\mathbb{Q}_p$? Does there exist algebraic number in ...
41 views

53 views

### Genus of extension $\mathbb{C}(T)(\sqrt{T^n + 1})$

Let $k = \mathbb{C}$ and $K$ is the extension $\mathbb{C}(T)(\sqrt{T^n + 1})$ of $\mathbb{C}(T)$ with $n \ge 2$ an even integer. I suspect that the genus of $K$ is $(n - 2)/2$, but all attempts at ...
104 views

59 views

### Irrational numbers in a cyclotomic extension $\mathbb{Q}(\zeta_N)$

Let $\zeta_N$ be a primitive root of unity so that $\mathbb{Q}(\zeta_N)$ is an $n$-th cyclotomic extension. I am interested in irrational (real) numbers $\alpha$ in $\mathbb{Q}(\zeta_N)$ such that ...
77 views

### Kummer map and cohomology group for an elliptic curve

Let $E=E_q$ be the Tate ellipitc curve over a finite extension $K$ of $\mathbb{Q}_p$ for a $q$. Let $T$ be its p-adic Tate module. Let $\mathfrak m$ be the maximal ideal in $K$. I saw in this paper ...
121 views

### Ideal $\mathfrak p^i$ is not principal

Let c be a postive squarefree integer. Let $K = \mathbb Q(\sqrt{-c})$. Let $p$ be a prime that splits in $K$ and let $\mathfrak p$ be a prime ideal above $p$. I need to prove the following: Prove ...
66 views

### How is the $p$-adic Tate module of a formal group defined?

I am familiar with the definition of the $p$-adic Tate module of an elliptic curve defined over a $p$-adic field $k$ (a finite extension of $\mathbb{Q}_p$). But I have also seen some instances where ...
13 views

### Close points with bounded minimal polynomial

Let $F$ be an ordered field, let $F'$ denote its real closure. Let $d$ be a natural number and let $A$ be a bounded subset of $F'$, all of whose elements are of degree $d$ over $F$. Is there a map ...
37 views

### Absolute values on $\mathbb{R}$ [closed]

Two related questions: a) is there some characterization of all the absolute values on $\mathbb{R}$? (similar to Ostrowski for $\mathbb{Q}$) b) are there non Archimedean absolute values on ...
49 views

58 views

### Using quadratic reciprocity to motivate higher reciprocity laws?

I'm an undergraduate following Neukirch's Algebraic Number Theory; Please do not assume much more than chapters $1$ and $2$ of this book to answer. The topics covered are: algebraic number fields, ...
I’m looking at the sum of m-th powers of consecutive integers i.e. $Sm(p) = 1^m + 2^m +…+(p-1)^m$ I need to prove: Let $p$ be an odd prime. Prove the following congruences: \begin{align} Sm(p^2) ...