Identical Zero Sets for two distinct irreducible polynomials In these notes on page 13, question number 9 asks to give example of two irreducible polynomials in $\mathbb{R}[X,Y]$ with identical zero sets. I can think of trivial examples like $x^2+y^2$ and $x^4+y^2$, both of which vanish only on the origin. Are there non-trivial examples where the zero sets are dimension $1$, for example?

The answer is no. This is simply Bezout's Theorem. A proof may be found in Michael Artin's Algebra.
