Irreducibility of polynomials in several variables

Question

Let $f(x_1,x_2,x_3)=x_1x_2+x_2x_3+x_3x_1\in k[x_1,x_2,x_3]$. Suppose $k$ a field with infinitely many elements. Is $f$ irreducible? Is there a standard way to prove this?

Answer 1

It is irreducible, which you can see by looking at the degree. That is a degree $2$ polynomial, for it to be reducible it must be the product of two degree $1$ polynomials. The leading term with lexicographic order is $x_1x_2$ so the leading terms of these polynomials must be $x_1$ and $x_2$. Now just take two general polynomials of this form: $$x_1 + ax_2 + bx_3 + c \qquad \text{and} \qquad x_2 + dx_3 + e$$ multiply them together and set the result equal to $f$. The coefficients give you a system of equations which you will soon find is inconsistant, there are no possible solutions for $a, b, c, d, e$.

p.s. You will not need to assume that the field is infinite.