Prove : If $I=(p(x))$ is a prime ideal in $ F[x]$ then $p(x)$ is irreducible.my claim is F need not to be field.
53367
In a principal ideal domain, prove that every non trivial prime ideal is a maximal ideal. What could be wrong in this approach?
87101
Irreducible but not prime in $K[X,Y]/(X^2-Y^3)$
17670
If $p\in R[X_1,\dots,X_n]$ is irreducible, is it still irreducible in $R[X_1,\dots,X_n,\dots,X_N]$?
137858
Showing that $\mathbb{Z}[i]/I$ is a finite field whenever $I$ is a prime ideal, and also finding its cardinality?
112427
Irreducibility of $p(x)=x^4-4x^2+8x+2$ over $\mathbb{Q}(\sqrt{-2})$- Dummit Foote Abstract algebra $9.4.10$
111114
Prove that an ideal in $k[x,y]$ is a prime ideal
110022
Showing prime and maximal ideal
132665
Show $p(X)$ (over a field) is irreducible iff $p(X+a)$ is irreducible
72230
Equivalence of definitions of prime ideal in commutative ring
85096
Let $K$ be a field and $f(x)\in K[x]$. Prove that $K[x]/(f(x))$ is a field if and only if $f(x)$ is irreducible in $K[x]$.
18186
Prove that ideal $I = \langle X^2,X+1\rangle$ is principal or not (in $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$ )
64878
Is this polynomial irreducible in $\mathbb{Q}$?
104486
Ideal in $\mathbb Z[x]$ which is not two-generated
112045
A property of minimal prime ideal
78143
Proving $\mathbb{F}_p/\langle f(x)\rangle$ with $f(x)$ irreducible of degree $n$ is a field with $p^n$ elements
46292
Find all primes $p>2$ for which $x^2+x+1$ is irreducible in $\mathbb{F}_p[x]$
135586
Prove that the preimage of a prime ideal is also prime.