query: Prime ideals in $F[x]$ when $F$ is a field
positive_passages.docid: 94174
positive_passages.text: Ring of polynomials over a field has infinitely many primes
negative_passages.docid: 80397
negative_passages.text: Show that this set of polynomials is ideal in F[x]
negative_passages.docid: 135586
negative_passages.text: Prove that the preimage of a prime ideal is also prime.
negative_passages.docid: 74558
negative_passages.text: Localization of ideals at all primes
negative_passages.docid: 3604
negative_passages.text: If F is a field, then $F[x,y]$ is a Principal Ideal Domain?
negative_passages.docid: 81221
negative_passages.text: The prime number theorem over a finite field - Lang's *Algebra*, Chapter V, Exercise 23(b)
negative_passages.docid: 149747
negative_passages.text: Prove that if I is a prime ideal of R then I[x] is a prime ideal of R[x]
negative_passages.docid: 135682
negative_passages.text: Are distinct prime ideals in a ring always coprime? If not, then when are they?
negative_passages.docid: 6699
negative_passages.text: An ideal that is maximal among non-finitely generated ideals is prime.
negative_passages.docid: 70751
negative_passages.text: A commutative ring whose all proper ideals are prime is a field.
negative_passages.docid: 155745
negative_passages.text: Ideals of $\mathbb{Z}[X]$
negative_passages.docid: 85206
negative_passages.text: Verifying proof :an Ideal $P$ is prime Ideal if $R/P$ is an integral domain.
negative_passages.docid: 104486
negative_passages.text: Ideal in $\mathbb Z[x]$ which is not two-generated
negative_passages.docid: 5360
negative_passages.text: Ideal contained in a finite union of prime ideals
negative_passages.docid: 137858
negative_passages.text: Showing that $\mathbb{Z}[i]/I$ is a finite field whenever $I$ is a prime ideal, and also finding its cardinality?
negative_passages.docid: 75558
negative_passages.text: Counter-example for prime ideals in a non commutative ring.
negative_passages.docid: 106550
negative_passages.text: The bijection between homogeneous prime ideals of $S_f$ and prime ideals of $(S_f)_0$