Tagged Questions
6
votes
1answer
270 views
Deciding whether $2^{\sqrt2}$ is irrational/transcendental
Is $2^\sqrt{2}$ irrational? Is it transcendental?
2
votes
1answer
103 views
Is there a rational univariat polynomial of degree 3 with 3 irrational roots?
The title pretty much asks my question: Does $f\in\mathbb{Q}[x]$ such that
$$ f(x)=(x-\alpha_1)(x-\alpha_2)(x-\alpha_3),$$
where $\alpha_1, \alpha_2, \alpha_3\in\mathbb{R}\setminus\mathbb{Q}\ $ ...
0
votes
3answers
673 views
Show that the ring of all rational numbers, which when written in simplest form has an odd denominator, is a principal ideal domain.
Show that the ring of all rational numbers $m/n$ with $n$ an odd integer is a principal ideal domain.
We haven't really discussed principal ideal domains. I've heard that this is easy, but I just ...
2
votes
1answer
92 views
Why is the group of rational numbers with odd denominators residually finite?
I want to prove that the additive group of rational numbers with odd denominators is residually finite. However, despite pondering for quite some time, I can't even find a single subgroup of finite ...
1
vote
1answer
52 views
How many sieves are there on a given rational number $q$?
Consider the poset category $\mathbb{Q}^{op}$, i.e. where $p \rightarrow q$ iff $p \geq q$. Take any $q \in \mathbb{Q}$. Then how many sieves are there for $q$ in $\mathbb{Q}^{op}$?
Supposedly, the ...
