query: Prove $R$ is a field.
positive_passages.docid: 24247
positive_passages.text: Ring Inside an Algebraic Field Extension
negative_passages.docid: 83100
negative_passages.text: Prove that the center of a ring is a subring.
negative_passages.docid: 13656
negative_passages.text: Prove existence of a real root.
negative_passages.docid: 61428
negative_passages.text: Show that an algebraically closed field must be infinite.
negative_passages.docid: 127002
negative_passages.text: Prove that for a nonzero element $a$ of a field $F$ with $q$ elements $a^{-1} = a^{q-2}$
negative_passages.docid: 62757
negative_passages.text: Showing that a ring is a field as well for one of the provided choices.
negative_passages.docid: 4706
negative_passages.text: Prove $f=x^p-a$ either irreducible or has a root. (arbitrary characteristic) (without using the field norm)
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: 46254
negative_passages.text: Why can't the Polynomial Ring be a Field?
negative_passages.docid: 49150
negative_passages.text: Subscript R notation for the finite fields
negative_passages.docid: 39114
negative_passages.text: Prove that a set in $\mathbb R^3$ is not an algebraic set
negative_passages.docid: 85096
negative_passages.text: Let $K$ be a field and $f(x)\in K[x]$. Prove that $K[x]/(f(x))$ is a field if and only if $f(x)$ is irreducible in $K[x]$.
negative_passages.docid: 76813
negative_passages.text: The ring $ℤ/nℤ$ is a field if and only if $n$ is prime
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: 10111
negative_passages.text: Prove or disprove that Q[√2] is a field
negative_passages.docid: 148192
negative_passages.text: In a field that is the direct product of two fields, why isn't it a field?