How do we check that a polynomial is irreducible in $\mathbb{C}[x,y]$

Question

we know that $\mathbb{C}$ is an algebraically closed field so every polynomial $f$ in $\mathbb{C}[x]$ can be represented as

$f=(x-a_{1})^{\alpha_{1}} ....(x-a_{n})^{\alpha_{n}} $

and the only irreducible polynomials are the constant polynomials and of polynomials of the form $g=x-a $ ,$a \in \mathbb{C} $

Let $f \in \mathbb{C}[x,y]$ be a polynomial

When do we say that $f$ is irreducible in $\mathbb{C}[x,y]$? Do the irreducible polynomials have an exact form like in the case of $\mathbb{C}[x]$? if not how we can check whether $f$ is irreducible or not ?

For example is this polynomial irreducible in $\mathbb{C}[x,y]$ or not

$h(x, y)=x^{p-1}+x^{p-2}+....+x+1-y$

Answer 1

The definition of irreducible polynomials $\mathbb{C}[x,y]$ is the same as in general commutative rings: A polynomial $f$ is irreducible if

• it is not a unit or $0$, i.e. constant in the case $\mathbb{C}[x,y]$
• if there is a factorization $f=ab$ then $a$ or $b$ has to be a unit.

I don't know if there is some kind of exact form for irreducible polynomials over $\mathbb{C}[x,y]$, the generalization of the fact that the irreducibles in $\mathbb{C}[x]$ are exactly those of the form $x-a$ is the Nullstellensatz concerning maximal ideals, but not single polynomials.

For your example: $h(x,y)=x^{p-1}+\dots+x+1-y$. This is irreducible since it has degree 1 in $y$.