149 views

94 views

Sum of 5 square roots equals another square root. What is the minimum possible value of the summed square root?

In the equation $\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{d}+\sqrt{e}=\sqrt{f}$, each variable is a distinct positive integer. What is the least possible value for $f?$ Out of purely trial and error, I have ...
120 views

A question about dense subsets of Euclidean spaces and Hilbert space

Let $S$ be the Euclidean plane and let $p(S)$ be a fixed point of $S$. Does there exist a dense subset $D(S)$ of $S$, such that no pair of distinct points of $D(S)$ are at the same distance from $p(S)$...
115 views

Prove that $[ \mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5}):\mathbb{Q}]=8.$

I have to solve the following exercise: Compute $[\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5}):\mathbb{Q}]$ and $\operatorname{Gal}(\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})/\mathbb{Q}).$ Here my attempt: ...
141 views

225 views

Sum of square root of non perfect square positive integers is always irrational?

Let $S$ be a set of positive integers such that no element of $S$ is a perfect square. Is it true that $\sum_{s_i \in S} \sqrt{s_i}$ is always irrational? Motivation. Suppose the length of the ...
211 views

Dimension of the algebraic closure of a continuum field of characteristic zero

Let me start by saying that I have no idea in algebra/number theory/whatever, so, please, forgive my ignorance. Let $\mathbb{F}$ be a field of characteristic zero and continuum cardinality, which ...
187 views

Prove that $[\mathbb{Q}(\sqrt[r]{p_1},\cdots ,\sqrt[r]{p_n}):\mathbb{Q}]=r^n$

We have $n$ distinct prime numbers $p_1,\cdots ,p_n$ and I am asked to show that $[\mathbb{Q}(\sqrt[r]{p_1},\cdots ,\sqrt[r]{p_n}):\mathbb{Q}]=r^n$ where $r\in \mathbb{N}$. I tried to solve it by ...
112 views

Technique for showing an element is not in a field?

I have an extension $\mathbb{Q}(5^{1/4}, i)$, and I want to show that $4^{1/4}$ is not contained in it. (I hope what I am trying to prove is true!) Anyways, my natural starting point is to assume ...
88 views

Irrationality of $(a_1+\sqrt{b_1})(a_2+\sqrt{b_2})$

Sorry, for a rather silly question. Suppose $a_1$, $b_1$, $a_2$, $b_2$ are integers, all different from zero, while $b_1$ and $b_2$ are co-prime positive integers, neither being a complete square. ...
Consider $x_1, x_2, ..., x_n \in \mathbb{R}$. We have to prove that each $\sqrt x$ is rational if the sum of $\sqrt x_1 + \ldots + \sqrt x_n$ is rational. I think that I could prove it using ...