Question: Find irreducible $f,g \in \mathbb{R}[x,y]$ such that $V(f) = V(g) \neq 0$ with the added requirement $f \neq \lambda g$ for $\lambda \in \mathbb{R} - \{0\}$.
Attempt: I think $f(x,y) = x^2 + y^2$ and $g(x,y) = x^4 + y^4$ will do the job but I am struggling to find a nice way to show $x^4 + y^4$ is irreducible over $\mathbb{R}$.
Is there an easier pair to take? Or a good way to see that this irreducible which I am missing?
The problem with $x^4+y^4$ is that it is homogenous, so it's basically the same as $(\frac{x}{y})^4+1$, which is not irreducible over $\mathbb{R}$.
If you take $y^2+x^4$, then it is irreducible over $\mathbb{R}(x)[y]$ since if some rational function $p(x)$ were a root for it when considered as a polynomial in $y$, then $p(x)^2=-x^4$, so in reduced form $p(x)$ is a polynomial, but then the leading coefficient of $p$ squared is the leading coefficient of $-x^4$, which is -1. Since the coefficients are in $\mathbb{R}$ this is impossible. Thus the quadratic has no roots as a polynomial in $y$ over $\mathbb{R}(x)[y]$, so it is irreducible in this ring and hence irreducible in $\mathbb{R}[x,y]$ as desired.
