query: If $F\subset D \subset E$ then $D$ must be a field
positive_passages.docid: 7828
positive_passages.text: Proof that an integral domain that is a finite-dimensional $F$-vector space is in fact a field
negative_passages.docid: 79725
negative_passages.text: Fields of arbitrary cardinality
negative_passages.docid: 29626
negative_passages.text: Is there a proper subfield $K\subset \mathbb R$ such that $[\mathbb R:K]$ is finite?
negative_passages.docid: 143219
negative_passages.text: If a field $F$ is an algebraic extension of a field $K$ then $(F:K)=(F(x):K(x))$
negative_passages.docid: 103654
negative_passages.text: A finite field extension that is not simple
negative_passages.docid: 76496
negative_passages.text: Showing that every field is an integral domain.
negative_passages.docid: 115449
negative_passages.text: Inclusion of Fields whose order is a prime power
negative_passages.docid: 131943
negative_passages.text: How to directly prove that $M$ is maximal ideal of $A$ iff $A/M$ is a field?
negative_passages.docid: 6855
negative_passages.text: Subfields of finite fields
negative_passages.docid: 17212
negative_passages.text: The preimage of a subset
negative_passages.docid: 124511
negative_passages.text: The set of algebraic numbers is a field.
negative_passages.docid: 136770
negative_passages.text: Any field that contains the integers contains the rationals as a subfield (Analysis proof)
negative_passages.docid: 144245
negative_passages.text: Shorter proof of $R/I$ is a field if and only if $I$ is maximal
negative_passages.docid: 94365
negative_passages.text: How to prove that if $D$ is countable, then $f(D)$ is either finite or countable?
negative_passages.docid: 70776
negative_passages.text: Characteristic of a field is $0$ or prime
negative_passages.docid: 79568
negative_passages.text: Cardinality of the set $D$
negative_passages.docid: 111347
negative_passages.text: Finding an example of field extensions satisfying certain properties