query: An infinite extension of $\mathbb Q$
positive_passages.docid: 27784
positive_passages.text: Proving that $\left(\mathbb Q[\sqrt p_1,\dots,\sqrt p_n]:\mathbb Q\right)=2^n$ for distinct primes $p_i$.
negative_passages.docid: 77548
negative_passages.text: Given an arbitrary infinite set $S$, how to construct an injection $\mathbb{N} \to S$ with the Axiom of Choice?
negative_passages.docid: 100134
negative_passages.text: Proving existence of a surjection $2^{\aleph_0} \to \aleph_1$ without AC
negative_passages.docid: 72144
negative_passages.text: Show that $\mathbb Q(\sqrt p) \not\simeq\mathbb Q(\sqrt q)$
negative_passages.docid: 42810
negative_passages.text: Is $(\mathbb{Q},+)$ isomorphic to $(\mathbb{Q}^n,+)$?
negative_passages.docid: 79778
negative_passages.text: $\kappa\cdot\kappa= \kappa,$ for infinite cardinals
negative_passages.docid: 76501
negative_passages.text: How can we show that $\mathbb Q$ is not a free $\mathbb Z$-module?
negative_passages.docid: 139078
negative_passages.text: Infinite partition of $\mathbb N$ by infinite subsets
negative_passages.docid: 24924
negative_passages.text: Computing an infinite product
negative_passages.docid: 103654
negative_passages.text: A finite field extension that is not simple
negative_passages.docid: 132868
negative_passages.text: Bijection between an infinite set and its union of a countably infinite set
negative_passages.docid: 97878
negative_passages.text: Are there uncountably many surjections from $\mathbb N$ to $\mathbb N$?
negative_passages.docid: 35379
negative_passages.text: Spectrum of infinite product of rings
negative_passages.docid: 155125
negative_passages.text: Every ordered field has a subfield isomorphic to $\mathbb Q$?
negative_passages.docid: 156006
negative_passages.text: Cardinality of set containing all infinite subsets of $\mathbb{Q}$
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: 41035
negative_passages.text: Mapping that proves $\mathbb{Z} \subset \mathbb{Q}$