query: Field Show that F is a field under + and $\cdot$ and F = $\{a + b\sqrt{3}\} | a, b \in \mathbb Q\}$
positive_passages.docid: 98577
positive_passages.text: Show that the set $\mathbb{Q}(\sqrt{p})=\{a+b\sqrt{p}; a,b,p\in\mathbb{Q},\sqrt{p}\notin \mathbb{Q}\}$ is a field
negative_passages.docid: 63478
negative_passages.text: Is $\mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}(\sqrt{3})$?
negative_passages.docid: 50172
negative_passages.text: Show that $\mathbb{Q}(\sqrt{2})$ is the smallest subfield of $\mathbb{C}$ that contains $\sqrt{2}$
negative_passages.docid: 138501
negative_passages.text: Is $\mathbf{Q}(\sqrt{2},\sqrt{3}) = \mathbf{Q}(\sqrt{6})$?
negative_passages.docid: 26207
negative_passages.text: If $f$ is a quadratic and $f(x)>0\;\forall x$, and $g= f + f' + f''$, prove $g(x)>0\; \forall x$
negative_passages.docid: 124564
negative_passages.text: Prove the ring $a+b\sqrt[3]{2}+c\sqrt[3]{4}$ has inverse and is a field
negative_passages.docid: 155165
negative_passages.text: Prove that $\mathbb F_8=\mathbb F_2[X]/(X^3+X+1)$
negative_passages.docid: 72144
negative_passages.text: Show that $\mathbb Q(\sqrt p) \not\simeq\mathbb Q(\sqrt q)$
negative_passages.docid: 93328
negative_passages.text: $\mathbb{F}_p[X]/(X^2+X+1)$ is a field iff $p \equiv 2 \bmod 3$
negative_passages.docid: 73412
negative_passages.text: Splitting field of $x^6+x^3+1$ over $\mathbb{Q}$
negative_passages.docid: 127764
negative_passages.text: Showing that $\mathbb{Q}[\sqrt{2}, \sqrt{3}]$ contains multiplicative inverses
negative_passages.docid: 136770
negative_passages.text: Any field that contains the integers contains the rationals as a subfield (Analysis proof)
negative_passages.docid: 56420
negative_passages.text: Is $\mathbb{Q}(\sqrt[3]{3}, \sqrt[4]{3})$ a Galois extension of $\mathbb{Q}$
negative_passages.docid: 69219
negative_passages.text: Prove there is a point $z\in[a,b]$ at which $f(z)=\frac{f(x_1)+f(x_2)+\cdots+f(x_k)}{k}$
negative_passages.docid: 19457
negative_passages.text: Show that: $\inf(A+B) = \inf(A)+ \inf(B)$
negative_passages.docid: 62254
negative_passages.text: Show $F_{n+1} \cdot F_{n-1} = F_n^2 + (-1)^n$ for all $n \in \mathbb{N}$
negative_passages.docid: 83009
negative_passages.text: Showing that $R(x)$ is a proper subset of $R((x))$ if $R$ is a field