Finding a polynomial that generates an ideal of a polynomial ring

Question

Let $a,b \in \mathbb{R}[x], a = x^5 - x^3 + 2x^2 - x, b = x^5 - x^4 - 8x + 5$. Let $I$ be the ideal in $\mathbb{R}[x]$ generated by a and b. Find a polynomial $p$, with $p \in \mathbb{R}[x]$ and $I = \mathbb{R}[x]p$

As $I$ is generated by a and b, I first thought about the gcd of a and b (euclidean algorithm). Such a $p$ clearly is $\in \mathbb{R}[x]$, but does the second condition, $I = \mathbb{R}[x]p$, applies for the gcd as well?

Am I right (and if yes, how could this be proven formally)? Is there an easier way to find such a $p$?

Answer 1

Hint: Every ideal in $\mathbb{R}[ x ]$ is principal.