query: Irreducible is prime in the ring $\mathbb{Z} + X\, \mathbb{Q}[X]$
positive_passages.docid: 158182
positive_passages.text: Let $R = \mathbb{Z} + x\mathbb{Q}[x]$. Find all the irreducibles in $R$.
negative_passages.docid: 134701
negative_passages.text: Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$
negative_passages.docid: 140808
negative_passages.text: Problem on Galois theory and irreducible polynomial
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: 135102
negative_passages.text: How do the definitions of "irreducible" and "prime" elements differ?
negative_passages.docid: 93291
negative_passages.text: Method for determining irreducibles and factorising in $\mathbb Z[\sqrt{d}]$
negative_passages.docid: 38364
negative_passages.text: Showing that $x^n -2$ is irreducible in $\mathbb{Q}[X]$
negative_passages.docid: 124926
negative_passages.text: Are Hilbert primes also Hilbert irreducible ? Furthermore, are Hilbert primes also primes in $\mathbb{ Z}$?
negative_passages.docid: 57884
negative_passages.text: Number of monic irreducible polynomials of prime degree $p$ over finite fields
negative_passages.docid: 114480
negative_passages.text: Example of a domain where all irreducibles are primes and that is not a GCD domain
negative_passages.docid: 134726
negative_passages.text: Show the polynomial is irreducible in $\textbf Z_2$
negative_passages.docid: 32391
negative_passages.text: Show that $x^2+x+4$ is irreducible over $\mathbb Z_{11}$.
negative_passages.docid: 121479
negative_passages.text: Why is $X^4+1$ reducible over $\mathbb F_p$ with $p \geq 3,$ prime
negative_passages.docid: 77620
negative_passages.text: Irreducible polynomials over $\mathbb Q$ and $\mathbb Z $
negative_passages.docid: 121714
negative_passages.text: Show that the polynomial $X^5 + X^3 + \bar{1}$ in $(\mathbb{Z}/2\mathbb{Z}[X])$ is irreducible
negative_passages.docid: 126808
negative_passages.text: Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ is reducible over $\mathbb{Q}$?
negative_passages.docid: 29545
negative_passages.text: $[(x-a_1)(x-a_2) \cdots (x-a_n)]^2 +1$ is irreducible over $\mathbb Q$