Criterion for Irreducibility of a polynomial in several variables?

[GargleBlast42]

Is there any criterion for the irreducibility of a polynomial in several variables over an algebraically closed field (or specifically for the complex numbers)? For one variable, we know this is simply that only degree one polynomials are irreducible, is there anything similar for several variables?

 

[mrbohn1]

If you can show that some specialization of the variables gives an irreducible polynomial in one variable, then this implies that the multivariate polynomial is irreducible.

So suppose you have some polynomial in K[x_1,x_2,x_3,...], where K is a field, and the x_i are indeterminates. Then all you need to do is find elements: a_2, a_3, a_4,... in K such that substituting a_i for x_i (i>1) gives an irreducible polynomial in K[x_1].

 

[Hurkyl]

Also, every polynomial in x and y over C is a polynomial in x over C(y).

 

[GargleBlast42]

If you can show that some specialization of the variables gives an irreducible polynomial in one variable, then this implies that the multivariate polynomial is irreducible.

So suppose you have some polynomial in K[x_1,x_2,x_3,...], where K is a field, and the x_i are indeterminates. Then all you need to do is find elements: a_2, a_3, a_4,... in K such that substituting a_i for x_i (i>1) gives an irreducible polynomial in K[x_1].

I'm not really sure if this works. Take for instance $$x^2 y+y$$ over the reals. As a polynomial in x, it is irreducible for any y, but as a polynomial in x,y it is obviously reducible $$x^2 y+y= y(x^2+1)$$

 

[Hurkyl]

As a polynomial in x, it is irreducible for any y

I do believe the zero polynomial counts as reducible.

edit: true but irrelevant

 

[mrbohn1]

I don't think that the constant polynomial does count. I should have been clearer: this only works if you consider your multivariate polynomial as a polynomial in one variable over the field of functions in the other variables. In your example, the field would be R(y), and reducible in this case would refer only to the x variable.