query: If $f(x)$ is irreducible over $\mathbb{Z}$ then prove that $\langle f(x) \rangle$ is not maximal ideal
positive_passages.docid: 17991
positive_passages.text: Nonconstant polynomials do not generate maximal ideals in $\mathbb Z[x]$
negative_passages.docid: 68202
negative_passages.text: Must an ideal generated by an irreducible element be a maximal ideal?
negative_passages.docid: 111996
negative_passages.text: Maximal and Prime Ideal in Z
negative_passages.docid: 155745
negative_passages.text: Ideals of $\mathbb{Z}[X]$
negative_passages.docid: 20790
negative_passages.text: What are irreducibles in the ring $R$ of entire functions?
negative_passages.docid: 116943
negative_passages.text: Proof for maximal ideals in $\mathbb{Z}[x]$
negative_passages.docid: 137233
negative_passages.text: $f$ is irreducible $\iff$ $G$ act transitively on the roots
negative_passages.docid: 104782
negative_passages.text: Polynomial irreducible - maximal ideal
negative_passages.docid: 78455
negative_passages.text: Prove that a nonzero integer p is prime if and only if the ideal (p) is maximal in Z.
negative_passages.docid: 112150
negative_passages.text: Prove that the ideal $(x^2 + x+2)$ is maximal in $F_3[x]$
negative_passages.docid: 102422
negative_passages.text: Irreducibly of $x^5-x-1$ over $\mathbb{F}_3$
negative_passages.docid: 132665
negative_passages.text: Show $p(X)$ (over a field) is irreducible iff $p(X+a)$ is irreducible
negative_passages.docid: 66355
negative_passages.text: A sufficient condition for irreducibility of a polynomial in an extension field.
negative_passages.docid: 63207
negative_passages.text: Zorn's lemma and maximal ideals
negative_passages.docid: 138572
negative_passages.text: Is $(xy-1)$ a maximal ideal in $\mathbb C[x,y]$?
negative_passages.docid: 1720
negative_passages.text: Show that $\langle 2,x \rangle$ is not a principal ideal in $\mathbb Z [x]$
negative_passages.docid: 129371
negative_passages.text: Maximal ideals in the ring of real functions on $[0,1]$