I am trying to solve an exercise in which I have to provide an example of two radical ideals $I,J \subset \mathbb Z[x]$ such that their sum $I+J$ is not radical.
I don't know how to attack this problem, I've came up with two examples of radical ideals in this ring: if $J=\langle p \rangle$ for a prime integer, then it is easy to see that this ideal is radical. The same for any irreducible polynomial $f$ in $\mathbb Z[x]$, $I=\langle f \rangle$ is radical, however I couldn't prove that for any two ideals of this form, their sum $I+J$ is not radical. I would like to understand why the property of being radical doesn't necessarily carry to the sum of ideals and if someone could help me to find an example to illustrate this in the ring $\mathbb Z[x]$. Thanks in advance.
Pick for example $I = (p)$ with $p$ a prime in $\mathbb{Z}$ and $J = (f)$ with $f$ an irreducible polynomial.
For example $f = x^2 + p$. Then $x^2 = f - p \in I + J$ but $x \notin I +J$, namely $I+J$ is not radical.
The idea is that you want to use $p$ to cancel the constant term of $f$ (or some other terms) and make what remains a perfect power.
You can cook up examples of higher degree with the same strategy. For example using Eisenstein criterion, for every given prime $p$ you can build an example with an $f$ of arbitrarily high degree, by taking $f = x^n + p$ (or adding other terms all divisible by $p$).