query: Is $\sqrt{2}^\sqrt{2}$ rational?
positive_passages.docid: 19676
positive_passages.text: Is it known whether ${\sqrt{2}}^{\sqrt{2}}$ is irrational?
negative_passages.docid: 101835
negative_passages.text: Which rationals can be written as the sum of two rational squares?
negative_passages.docid: 106242
negative_passages.text: Why no common factors in proving root 2 is irrational?
negative_passages.docid: 91101
negative_passages.text: Proving $\sqrt2$ is irrational
negative_passages.docid: 97071
negative_passages.text: Is the cube root of a prime number rational?
negative_passages.docid: 59391
negative_passages.text: $p,q,r$ primes, $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is irrational.
negative_passages.docid: 158730
negative_passages.text: Is $\ln(\ln(n))$ irrational for any integer $n>1$?
negative_passages.docid: 32760
negative_passages.text: Is $\mathbf{Q}(\sqrt{2}, \sqrt{3}) = \mathbf{Q}(\sqrt{2}+\sqrt{3})$?
negative_passages.docid: 6275
negative_passages.text: How can you prove that the square root of two is irrational?
negative_passages.docid: 104300
negative_passages.text: Help justifying that $\mathbb Q(\sqrt[3]{2})$ is not a splitting field over $\mathbb Q$.
negative_passages.docid: 20315
negative_passages.text: $\sqrt[31]{12} +\sqrt[12]{31}$ is irrational
negative_passages.docid: 63409
negative_passages.text: If $x^2+ax+b=0$ has a rational root, show that it is in fact an integer
negative_passages.docid: 143054
negative_passages.text: Can you raise $\pi$ to a real power to make it rational?
negative_passages.docid: 75639
negative_passages.text: Showing that $\sqrt[3]{2}\notin\Bbb Q(\alpha_1,...,\alpha_k)$ where $\alpha_i^2\in\Bbb Q\ \forall i$
negative_passages.docid: 92226
negative_passages.text: Prove $\sqrt6$ is irrational
negative_passages.docid: 134765
negative_passages.text: Is an integer a sum of two rational squares iff it is a sum of two integer squares?
negative_passages.docid: 53331
negative_passages.text: $\mathbb{Q}(\sqrt2, \sqrt3)$ is not isomorphic to $\mathbb{Q}(\sqrt[4]2)$