query: $(x^a-y^b)$ is a prime ideal in $R[x,y]$ where $R$ is a domain and $\gcd(a,b)=1$.
positive_passages.docid: 2043
positive_passages.text: $X^n-Y^m$ is irreducible in $\Bbb{C}[X,Y]$ iff $\gcd(n,m)=1$
negative_passages.docid: 145630
negative_passages.text: What is the ideal class group of the ring $\mathbb{R}[x,y]/(x^2+y^2-1)$?
negative_passages.docid: 587
negative_passages.text: Ideal Generated by the Union of Two Ideals
negative_passages.docid: 50809
negative_passages.text: Proving $R-S$ contains a prime ideal when $S$ is a multiplicative set
negative_passages.docid: 84386
negative_passages.text: Domain strictly contained in the intersection of localizations at the primes of height one
negative_passages.docid: 87780
negative_passages.text: Is the ideal $(2,X+1)\subset\Bbb{Z}[X]$ prime, maximal or neither?
negative_passages.docid: 155745
negative_passages.text: Ideals of $\mathbb{Z}[X]$
negative_passages.docid: 138114
negative_passages.text: Proving the quotient of a principal ideal domain by a prime ideal is again a principal ideal domain
negative_passages.docid: 100961
negative_passages.text: Ideals in valuation rings
negative_passages.docid: 45962
negative_passages.text: Show that any prime ideal from such a ring is maximal.
negative_passages.docid: 72640
negative_passages.text: $(m,n) = (\gcd(m,n))$ as ideals in $\Bbb Z$
negative_passages.docid: 78380
negative_passages.text: Dedekind domain with a finite number of prime ideals is principal
negative_passages.docid: 75558
negative_passages.text: Counter-example for prime ideals in a non commutative ring.
negative_passages.docid: 82938
negative_passages.text: $\gcd(a,b)=1, x^a = y^b\Rightarrow x = n^b$, $ y = n^a$ for an integer $n$.
negative_passages.docid: 56800
negative_passages.text: Intersection of prime ideals
negative_passages.docid: 81296
negative_passages.text: If A and B are ideals of a ring, show that A + B = $\{a+b|a \in A, b \in B\}$ is an ideal
negative_passages.docid: 76507
negative_passages.text: $\gcd(b^x - 1, b^y - 1, b^ z- 1,\dots) = b^{\gcd(x, y, z,\dots)} -1$