query: Prove that $\langle f(x) \rangle $ is prime ideal in $\Bbb Z[x]$ if $f(x)$ is irreducible over $\Bbb Z$
positive_passages.docid: 152109
positive_passages.text: In a UFD if $x$ is irreducible, then $(x)$ is prime
negative_passages.docid: 147671
negative_passages.text: Find all ideals of $\mathbb R[x] / \langle x^2-3x+2\rangle$
negative_passages.docid: 130466
negative_passages.text: Proving every prime ideal is maximal in a finite dimensional ring over a field.
negative_passages.docid: 116430
negative_passages.text: If $I = \langle 2\rangle$, why is $I[x]$ not a maximal ideal of $\mathbb Z[x]$, even though $I$ is a maximal ideal of $\mathbb Z$?
negative_passages.docid: 74558
negative_passages.text: Localization of ideals at all primes
negative_passages.docid: 102871
negative_passages.text: $\langle r\rangle$ maximal $\iff r$ irreducible
negative_passages.docid: 67203
negative_passages.text: Complement of maximal multiplicative set is a prime ideal
negative_passages.docid: 72230
negative_passages.text: Equivalence of definitions of prime ideal in commutative ring
negative_passages.docid: 50809
negative_passages.text: Proving $R-S$ contains a prime ideal when $S$ is a multiplicative set
negative_passages.docid: 87103
negative_passages.text: Quick question about ideals of field $F$ in $F[x]$ are principal.
negative_passages.docid: 106550
negative_passages.text: The bijection between homogeneous prime ideals of $S_f$ and prime ideals of $(S_f)_0$
negative_passages.docid: 43230
negative_passages.text: Is $\langle1-i\rangle$ a maximal ideal in $\mathbb{Z}[i]$?
negative_passages.docid: 101637
negative_passages.text: Let $R$ be an integral domain. If $x \in R$ is prime, then $x$ is irreducible.
negative_passages.docid: 110022
negative_passages.text: Showing prime and maximal ideal
negative_passages.docid: 93658
negative_passages.text: Proving that an ideal in a PID is maximal if and only if it is generated by an irreducible
negative_passages.docid: 5360
negative_passages.text: Ideal contained in a finite union of prime ideals
negative_passages.docid: 250
negative_passages.text: An irreducible $f\in \mathbb{Z}[x]$, whose image in every $(\mathbb{Z}/p\mathbb{Z})[x]$ has a root?