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Prove that root of number is rational.

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 ...
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Why need open in the Baire Category Theorem

In the statement of the Baire Category Theorem, one needs to include openness of a set so that the theorem holds. Question: what is the example such that countable intersection of dense sets is not ...
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What are the applications of the result that $x$ is not a square in $F(x)$?

Let $F(x)$ denote the field of quotients of the ring $F[x]$. Prove that there is no element in $F(x)$ whose square is $x$. Solution:If possible let $\frac{f(x)}{g(x)} \in F(x),g(x)\neq 0$ such ...
Is $\mathbb Q(\sqrt 2) \times \mathbb Q(\sqrt 3)=\mathbb Q(\sqrt 2,\sqrt 3)$ if I prove $\sqrt 2,\sqrt 3$ are L.I. over $\mathbb Q$? [duplicate]
I proved that $\{1,\sqrt 2\}$ and $\{1,\sqrt 3\}$ are respective bases of $\mathbb Q(\sqrt 2)$ and $\mathbb Q(\sqrt 3)$ over $\mathbb Q$. I want to show in some sense that since $\sqrt 2,\sqrt 3$ are ...