Intereting Posts

Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power
(Diophantine?) Equations With Multiple Variables
Counting square free numbers co-prime to $m$
An algorithm for making conditionally convergent series take arbitrary values?
Integral ${\large\int}_0^{\pi/2}\frac{\sin\left(x-a\ln2\cdot\tan x\right)}{\left(e^{2\pi a\tan x}-1\right)\cdot\cos x}\,dx$
Inverse of a function's integral
How to calculate $\lim_{n\to\infty}(1+1/n^2)(1+2/n^2)\cdots(1+n/n^2)$?
Are there more general spaces than Euclidean spaces to have the Heine–Borel property?
How to solve this sequence $165,195,255,285,345,x$
Fourier Analysis textbook recommendation
prove that $A(n) : \left(\frac n3\right)^n\lt n!\lt \left(\frac{n}{2}\right)^n$ for all $n\ge 6$
finite dimensional range implies compact operator
Prove that CX and CY are perpendicular
On Martingale betting system
A collection of pairwise disjoint open intervals must be countable

Let $K$ be a finite extension of $\mathbb{Q}$ (a number field) and $\mathcal{O}_K$ its ring of integers. One defines the norm of an element $\alpha\in K$ to be the determinant of the transformation $m_\alpha: K\to K$ of multiplication by $\alpha$ (where $K$ is considered as a vector space over $\mathbb{Q}$).

Now sometimes the integer ring is also a euclidean domain, i.e. it has a “euclidean norm” satisfying the defining property of division algorithm. My question is: in an integer ring which is also euclidean, will the norm defined above also serve as a euclidean norm?

Put otherwise: is there an example for a euclidean integer ring whose norm is *not* an euclidean norm?

- Definition of Simple Group
- Are finitely generated projective modules free over the total ring of fractions?
- Show $\langle a^m \rangle \cap \langle a^n \rangle = \langle a^{\operatorname{lcm}(m, n)}\rangle$
- Is the ideal generated by irreducible element in principal ideal domain maximal?
- If G is a finite abelian group and $a_1,…,a_n$ are all its elements, show that $x=a_1a_2a_3…a_n$must satisfy $x^2=e$.
- Isomorphism between the group $(\mathbb Z, +)$ and $(\mathbb Q_{>0}, .)$

- What do prime ideals in $k$ look like?
- Showing that $x^n -2$ is irreducible in $\mathbb{Q}$
- $\mathbb{Z}/(X^2-1)$ is not isomorphic with $\mathbb{Z}\times \mathbb{Z}$
- unique factorization of matrices
- “The Egg:” Bizarre behavior of the roots of a family of polynomials.
- Why is $\gcd(a,b)=\gcd(b,r)$ when $a = qb + r$?
- If a ring element is right-invertible, but not left-invertible, then it has infinitely many right-inverses.
- Show $A_n$ has no subgroups of index 2
- Question about Subrings and integraly closed
- Similar matrices and field extensions

Wikipedia says that “$\mathbb Q(\sqrt {69})$ is Euclidean but not norm-Euclidean. Finding all such fields is a major open problem, particularly in the quadratic case.”

- With infinite size, we can have $P \cdot M = M \cdot D $ (D diagonal) but where $M^{-1}$ does not exist. Can we say “P is diagonalizable”?
- Prove that $p^j q^i$ cannot be a perfect number for $p, q$ odd, distinct primes.
- Riccati differential equation $y'=x^2+y^2$
- sum of a series
- Show that $\sum_{k=0}^n\binom{3n}{3k}=\frac{8^n+2(-1)^n}{3}$
- Find the value of : $\lim\limits_{n\to \infty} \sqrt {\frac{(3n)!}{n!(2n+1)!}} $
- Find volume of the cap of a sphere of radius R with thickness h
- Why is $PGL(2,4)$ isomorphic to $A_5$
- An unexpected application of non-trivial combinatorics
- How to find this minimum of the value
- Determining whether the extremal problem has a weak minimum or strong minimum or both
- Topological invariants
- $A\subset \mathbb{R}$ with more than one element and $A/ \{a\}$ is compact for a fixed $a\in A$
- Randomly dropping needles in a circle?
- Simple inequality for measures