Prove that some bivariate polynomial is irreducible

Question

How can I prove that $p(x,y)=2xy+x^2+y^3$ is irreducible over the ring $\mathbb{R}[x,y]$? Is there, for instance, any way to generalize Eisenstein's criterion to polynomials in several variables?

I have tried a proof by contradiction without success, also I am looking for stronger techniques when it comes to polynomials in more than one variable. Any help?

Answer 1

Yes, you can use the Eisenstein's criterion in this general form:

Given an integral domain $D$, let $f=\sum_{i=0}^n a_iz^i$ be an element of $D[z]$. Suppose there exists a prime ideal $\mathfrak p$ of $D$ such that
$a_i ∈\mathfrak p$ for each $i ≠ n$,
$a_n ∉\mathfrak p$, and
$a_0 ∉\mathfrak p^2$.
Then $f$ cannot be written as a product of two non-constant polynomials in $D[z]$. If in addition $f$ is primitive (i.e., it has no non-trivial constant divisors), then it is irreducible in $D[z]$.

In your case write $x^2+2xy+y^3=(x+y)^2-y^2+y^3$, set $z=x+y$, $D=\mathbb R[y]$ and $\mathfrak p=(y-1)$. (Maybe it's helpful to write the polynomial $z^2+y^2(y-1)$, and thus can identify $a_0=y^2(y-1)$, and $a_1=1$.)

Answer 2

Hint: $y^3+2xy+x^2 = (y+P(x))(y^2+ Q(x)y + R(x))$, then what are the values of $P,Q,R$?

Answer 3

Hints :

For $y=2$ you have $p(x,2)=x^2+4x+8$ which is irreducible..

So.... Can you guess something???