# Tagged Questions

17 views

### Euclidean Algorithm in $\mathbb{Z}[w], w=\dfrac{1+\sqrt{-7}}{2}$

We are in the ring $\mathbb{Z}[w], w=\dfrac{1+\sqrt{-7}}{2}$. I am trying to find the gcd of 2-7 and 11. What I usually do is set up: 11=q(w-7) + r. I'll find q and r, then write: w-7=q(r)+r_new. ...
29 views

### gcd's in non-UFD rings

In a UFD ring we have that for coprime $a,b \in R$, i.e. $(a,b)=1$: $$a|cb \Rightarrow a|c$$ Does this property hold for non-UFD rings? I think not but do not recall a standard ...
102 views

### Roots of $x^n - 1$ in an algebraically closed field of prime characteristic

Let $F$ be an algebraically closed field of characteristic $p$ , and let $n$ be a positive integer. Consider $g := x^n - 1 \in F[x]$ Is it true that $g$ has distinct roots in $F$ if and only if ...
32 views

### Subring of Gaussian integers has no greatest common divisor property [duplicate]

Problem is: Produce elements a and b in the domain $R := \{x+2y\sqrt{-1} \mid x, y \in \mathbb{Z}\}$ having no gcd. How can produce this? Actually I use norm function, and brute force, but what ...