query: What is the cardinality of a transcendence basis of $\mathbb{C}$ over $\mathbb{Q}$?
positive_passages.docid: 155750
positive_passages.text: Why does $\mathbb{C}$ have transcendence degree $\mathfrak{c}$ over $\mathbb{Q}$?
negative_passages.docid: 35105
negative_passages.text: Density of ring of algebraic integers in $\mathbb C$
negative_passages.docid: 36325
negative_passages.text: Embedding ordinals in $\mathbb{Q}$
negative_passages.docid: 153652
negative_passages.text: Countable ordinals are embeddable in the rationals $\Bbb Q$ -- proofs and their use of AC
negative_passages.docid: 122947
negative_passages.text: What is the cardinality of the set of infinite cardinalities?
negative_passages.docid: 31173
negative_passages.text: How to show $(a^b)^c=a^{bc}$ for arbitrary cardinal numbers?
negative_passages.docid: 95240
negative_passages.text: Show that $({\mathbb{Q}},+)$ is not finitely generated using the Fundamental Theorem of Finitely Generated Abelian Groups.
negative_passages.docid: 27704
negative_passages.text: Cardinality of quotient ring $\mathbb{Z_6}[X]/(2X+4)$
negative_passages.docid: 43300
negative_passages.text: The degree of $\sqrt{2} + \sqrt[3]{5}$ over $\mathbb Q$
negative_passages.docid: 145140
negative_passages.text: Subfield of $\mathbb{Q}(\sqrt[n]{a})$
negative_passages.docid: 150064
negative_passages.text: Cardinality of "$x-y\in\Bbb Q$"-equivalence class of $1/\sqrt2$
negative_passages.docid: 112909
negative_passages.text: is a number field by definition a subfield of $ \mathbb C $?
negative_passages.docid: 148101
negative_passages.text: Basis for $\mathbb{Z}^2$
negative_passages.docid: 63390
negative_passages.text: Infinite algebraic extensions of $\mathbb{Q}$
negative_passages.docid: 42810
negative_passages.text: Is $(\mathbb{Q},+)$ isomorphic to $(\mathbb{Q}^n,+)$?
negative_passages.docid: 51687
negative_passages.text: Is the set $ \ (\mathbb{Z}_p \setminus \mathbb{Z}) \cap \mathbb{Q}$ non-empty?
negative_passages.docid: 98731
negative_passages.text: Show that there is no epimorphism from ($\mathbb{Q}$, +) to ($\mathbb{Z}$, +)