query: Prove that $\mathbb{Q}(\sqrt{a}, \sqrt{b})= \mathbb{Q}( \sqrt{a}+\sqrt{b})$
positive_passages.docid: 158161
positive_passages.text: $\mathbb{Q}(\sqrt p_{1}+\sqrt p_{2})=\mathbb{Q}(\sqrt p_{1},\sqrt p_{2})$ for $p_{1},p_{2}$ primes.
negative_passages.docid: 127296
negative_passages.text: prove that a|b and b|a iff a=+-b
negative_passages.docid: 109133
negative_passages.text: $a+b+c = 3$, prove that :$a\sqrt{a+3}+b\sqrt{b+3}+c\sqrt{c+3} \geq 6$
negative_passages.docid: 125164
negative_passages.text: How to prove the existence of $b$ in $Q$ such that $a<b^2<c$ in $Q$?
negative_passages.docid: 65218
negative_passages.text: Let $m \in \mathbb{Z^+} , n \in \mathbb{Z^+}$ and let $d=\gcd(m,n)$. Prove that $m\mathbb{Z}+n\mathbb{Z}=d\mathbb{Z}$
negative_passages.docid: 31196
negative_passages.text: Prove $\binom{n}{a}\binom{n-a}{b-a} = \binom{n}{b}\binom{b}{a}$
negative_passages.docid: 118981
negative_passages.text: Proving $\sqrt{ab} = \sqrt a\sqrt b$
negative_passages.docid: 68838
negative_passages.text: Why $\sqrt[3]{3}\not\in \mathbb{Q}(\sqrt[3]{2})$?
negative_passages.docid: 16163
negative_passages.text: Show $\mathbb{Q}(\sqrt{2},\sqrt{3})$ is a simple extension field of $\mathbb{Q}$.
negative_passages.docid: 38489
negative_passages.text: Degree of the extension $\mathbb{Q}(\sqrt{a+\sqrt{b}})$ over $\mathbb{Q}$
negative_passages.docid: 88425
negative_passages.text: Prove that $\,\sqrt [n] n < 1 + \sqrt{\frac{2}{n}}\,$
negative_passages.docid: 39779
negative_passages.text: If $a,b,c>0$ and $a+b+c=1$, prove that $\sqrt{\frac{ab}{c+ab}}+\sqrt{\frac{bc}{a+bc}}+\sqrt{\frac{ac}{b+ac}} \le \frac{3}{2}$.
negative_passages.docid: 98496
negative_passages.text: Let $a,b \in \mathbb{Q^{+}}$ then $\sqrt{a} + \sqrt{b} \in \mathbb{Q} $ iff $\sqrt{a} \in \mathbb{Q}$ and $\sqrt{b} \in \mathbb{Q}$
negative_passages.docid: 14162
negative_passages.text: How can I prove $\sup(A+B)=\sup A+\sup B$ if $A+B=\{a+b\mid a\in A, b\in B\}$
negative_passages.docid: 65297
negative_passages.text: Proof that for $a,b,c\in \mathbb{R}$, hold that $\left |\sqrt {a^2+b^2}-\sqrt {a^2+c^2} \right |\leq \left |b-c \right |$.
negative_passages.docid: 46565
negative_passages.text: Show that $\mathbb{Q}(\sqrt{2})$ is a field.
negative_passages.docid: 20878
negative_passages.text: If $(a,b)=1$ then prove $(a+b, ab)=1$.