Articles of algebraic number theory

How to make sense of the ideal norm in a localization?

This is a follow-up to this question: Localization of finite modules, or: compatibility of ideal norms with localization at a prime number Let $K$ be an algebraic number field and $N:\mathcal{I}(\mathcal{O}_K)\to \mathbb{Z}$ be $N(\mathfrak{a})=\# (\mathcal{O}_K/\mathfrak{a})$, where $\mathcal{I}(\mathcal{O}_K)$ denotes the set of non-zero ideals of $\mathcal{O}_K$. Let $S\subset \mathcal{O}_K$ be a multiplicative subset. Is there a […]

Norms and traces example

Example: Let $L=\mathbb{Q}(\sqrt{d})$ be a quadratic extention of $F=\mathbb{Q}$ with square-free integer $d$.Then, $g_{a+b\sqrt{d}}(X)=(X-a-b\sqrt{d})(X-a+b\sqrt{d})=X^2 -2aX+(a^2 -db^2),$ so, $Tr_{\mathbb{Q}(\sqrt{d})/\mathbb{Q}}(a+b\sqrt{d})=2a$, $N_{\mathbb{Q}(\sqrt{d})/\mathbb{Q}}(a+b\sqrt{d})=a^2-db^2$. In particular, an integer number $c$ is a sum of two squares iff $c\in N_{\mathbb{Q}(\sqrt{-1})/\mathbb{Q}}O_{\mathbb{Q}(\sqrt{-1})}$. More generally, $c$ is of the form $a^2 -db^2$ with $a,b\in \mathbb{Z}$ and square-free $d$ not congruent to $1 \mod 4$ […]

Generalized Hensel lifting in a Dedekind domain

The following theorem is known as generalized Hensel lifting(see here). Can we prove this without using $P$-adic completion? Theorem Let $A$ be a Dedekind domain. Let $P$ be a non-zero prime ideal of $A$. Let $f(x) \in A[x]$ be a polynomial. Let $a \in A$. Suppose $P^r||f'(a)$. Let $m = 2r + 1$. Let $k […]

Demonstrating that a particular ideal is maximal in a number field

I have three ideals, each with two elements of $\mathbb Z[\sqrt{-5}]$. If you show me how to show one of them is maximal (hence prime) then I think I can manage to do the remaining two by myself. $\langle 2, 1 + \sqrt{-5}\rangle$

Can $\theta$ be a rational multiple of $2\pi$?

In this question the author conjectured that a certain angle $\theta$ could be a rational multiple of $2\pi$ only in the case it was an integer multiple of $\pi/2$. In my answer I found an explicit representation for $\theta$: $$ \cos\theta=1+2\cos\left(2r\pi\right), $$ where $r$ is any rational number such that the right hand side of […]

Generalisation of Fermat's Little Theorem in Function Fields

There is a well-known generalisation of Fermat’s Little Theorem as for any $a\in\mathbb{Z}$ and $n\in\mathbb{N}$ we have $$\sum_{d\mid n} \mu(n/d) a^d\equiv 0\pmod n.$$ where $\mu$ is the Möbius function. I wonder if this formula still holds in function fields. Say $F$ be a finite field and $F[x]$ be the polynomial ring over $F$. Then $F[x]$ […]

The norm of an ideal and the norms of its elements

Let $F$ be a number field, $\mathfrak a$ a fractional ideal. If $\mathfrak a$ is a prime ideal in $\mathcal O_F$ lying over prime ideal $p\mathbb Z$ in $\mathbb Z$ then define its norm as $p^f$ where $f=deg[\mathcal O_F/\mathfrak a:\mathbb Z/p\mathbb Z]$. The norm continues to all fractional ideals by multiplicativity. Does the norm of […]

Classifying algebraic integers satisfying a positivity condition

Let $a $ be an algebraic integer such that $1/a$ is also an algebraic integer belonging to the ring of integers of $\mathbb {Q}(a) $. Then, what is the condition for $a $ to satisfy: For any integer-coefficient polynomial $f (x) $ and any Galois conjugate $a’$ of $a $, $f (a’)f (1/a’)\ge 0. $ […]

Multiples of 4 as sum or difference of 2 squares

Is it true that for $n \in \mathbb{N}$ we can have $4n = x^{2} + y^{2}$ or $4n = x^{2} – y^{2}$ for $x,y \in \mathbb{N} \cup (0)$. I was just working out a proof and this turns out to be true from $n=1$ to $n=20$. After that I didn’t try, but I would like […]

Finding an example of a non-rational p-adic number

We know that every rational number can be written as a $p$-adic integer with expansion $\sum\limits_{n=-m}^\infty a_n p^n$, where $a_n\in\{0,\dots,p-1\}$ and $m\in\mathbb{N}$; therefore there exists an injection $\mathbb{Q} \hookrightarrow \mathbb{Q}_p$. But how do I show that $\mathbb{Q}_p$ is bigger? How do I find an example of a $p$-adic number which is not rational? I heard […]