Articles of algebraic number theory

Class number/ quadratic field/divisibility-$2$

As I am poor in construction of mathematical problem, I am not getting good answers from MSE members. However, this time I constructed the following problem in best possible way. So, I hope I will get some good answers from readers! Let $n$ be a positive integer, $k$ be an odd prime number, and $e$ […]

Are the Eisenstein integers the ring of integers of some algebraic number field? Can this be generalised?

The Eisenstein integers are $\mathbb Z[\omega]$ where $\omega$ is the primitive third root of unity. If $K$ is some algebraic number field, can $\mathcal O_K$ be isomorphic to $\mathbb Z[\omega]$? Attempt: The rank of $\mathcal O_K$ as a $\mathbb Z$-module is the degree of the extension $K$ of $\mathbb Q$. I believe that $\mathbb Z[\omega]$ […]

How to factor the ideal $(65537)$ in $\mathbb Z$?

This question is related to How to factor ideals in a quadratic number field? In Algebraic Number Theory by W. Stein he makes a remark about the factorization of $65537$ in $\mathbb Z[i]$. I checked this in Sage and the result is different. What is an explanation of this difference?

Is $O_L$ a free $O_K$ module?

This question already has an answer here: Integral basis of an extension of number fields 1 answer

Estimate number of solutions in the Roth's theorem

There is a fundamental theorem in Diophantine approximation : For algebraic irrational $\alpha$ $$\displaystyle \left \lvert \alpha – \frac{p}{q} \right \rvert < \frac{1}{q^{2 + \epsilon}}$$ with $\epsilon>0$, has finitely many solutions. can we estimate number of solutions $N_{\alpha}(\epsilon)$? for instance : what is the upper bound of $N_{\sqrt{2}}(1)$? number of solutions for $\sqrt{2}$, with $\epsilon=1$. […]

Basis for rank $n$ ring containing $1$.

Suppose $L$ is a finite (separable?) extension of a number field $K$, $\mathcal{O}_K$ is the ring of integers of $K$, and $\mathcal{O}_L$ is the integral closure of $\mathcal{O}_K$ in $L$. How can one prove that there is a basis $\mathcal{B}$ of $\mathcal{O}_L$ over $\mathcal{O}_K$ containing $1$? Any $K$-linearly independent set can be extended to a […]

Factorization of $p\mathcal{O}_K$ in $\mathcal{O}_K$

Let $K=\mathbb{Q}(\alpha)$, where $\alpha$ is a root of $f(x)=x^3+x+1$. If $p$ is a rational prime. What can you say about factorization of $p\mathcal{O}_K$ in $\mathcal{O}_K$? I have this: The discriminant is -31 and the minkowski bound, $M_K \leq 1.57$. Then $N(\mathcal{A})=1$, $\mathcal{A}$ a ideal class. I can say anything about the question?

If a cyclotomic integer has (rational) prime norm, is it a prime element?

Let $p$ be a rational prime. Consider the ring of integers $\mathbb{Z}[\zeta_p] $ of the $p$-th cyclotomic field $\mathbb{Q}(\zeta_p)$. If the norm $N(\alpha)$ of $\alpha \in \mathbb{Z}[\zeta_p]$ is a rational prime, must $\alpha$ be a prime element of $\mathbb{Z}[\zeta_p] $? If it helps, I only need the case where $N(\alpha) \equiv 1$ mod $p$.

$X^n + X + 1$ reducible in $\mathbb{F}_2$

I was told that sometimes in characteristic 2 that $X^n + X + 1$ is reducible mod 2. What is the smallest $n$ where that is true?

Decomposition of a prime number $p \neq l$ in the quadratic subfield of a cyclotomic number field of an odd prime order $l$

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Let $A$ be the ring of algebraic integers in $K$. Let $G$ be the Galois group of $\mathbb{Q}(\zeta)/\mathbb{Q}$. $G$ is isomorphic to $(\mathbb{Z}/l\mathbb{Z})^*$. Hence $G$ is a cyclic group of order $l – […]