-2
votes
0answers
45 views
12
votes
1answer
149 views

Existence of rational sequence such that a polynomial is split over $\Bbb{Q}$

Does there exist a sequence $(a_n)_{n\in \Bbb{N}}$ of rationals such that for all $n\in \Bbb{N}$, $a_n\neq 0$ and the polynomial $a_0+a_1X+\cdots+a_nX^n$ is split over $\Bbb{Q}$? I was asked this ...
0
votes
1answer
37 views

Show that two field extensions are the same

Can you help me with showing that these two field extensions are the same: $\mathbb{Q}(\sqrt{3}, \sqrt[3]{5})$ $\mathbb{Q}(\sqrt{3} + b\sqrt[3]{5})$, where $b\neq0$ is any rational number. Thanks ...
2
votes
1answer
265 views

Is every injective rational function $f:\mathbb Q\to\mathbb Q$ a polynomial?

I thought this might be quite easy to show, and then realized that the tools I know from real analysis aren't going to help here. Suppose we have a rational function: $$ f(X)=\frac{P(X)}{Q(X)} $$ ...
6
votes
3answers
282 views

Additive group of rationals has no minimal generating set

In a comment to Arturo Magidin's answer to this question, Jack Schmidt says that the additive group of the rationals has no minimal generating set. Why does $(\mathbb{Q},+)$ have no minimal ...
4
votes
1answer
153 views

Algebraic structure of a set of Egyptian fractions of a positive rational?

It is said that every positive rational number can be represented by infinitely many Egyptian fractions (defined as the sum of distinct unit fractions). I am struggling to understand in a formal way, ...
0
votes
1answer
139 views

Properties of homomorphisms of the additive group of rationals

Let $f : (\mathbb{Q},+) \longrightarrow (\mathbb{Q},+)$ be a non-zero homomorphism. Can we conclude that $f$ is bijective (or, if that fails, that $f$ is injective or surjective)? Context The ...
6
votes
1answer
529 views

Deciding whether $2^{\sqrt2}$ is irrational/transcendental

Is $2^\sqrt{2}$ irrational? Is it transcendental?
2
votes
1answer
119 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
1k 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
129 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
57 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 ...