Definition: We say that a ring $R$ is an integral $k$-algebra if it is a $k$-algebra that is an integral domain. We say that a ring $R$ is an geometrically integral $k$-algebra if it is a $k$-algebra such that for every field extension $k'\supset k$, the ring $R\otimes_{k}k'$ is an integral $k'$-algebra.
The first example I want to find is an integral $k$-algebra $R$ such that $k$ is a perfect field and $R$ is not a geometrically integral.
Secondly, when $k$ is an imperfect field, I want to find an integral $k$-algebra $R$ such that for a field extension $k'\supset k$, the tensor product $k'\otimes_{k}R$ has nonzero nilpotent element.
How can I find such examples?
Thanks in advance.
For the first example, consider $\mathbb C$ as an $\mathbb R$-algebra. This is clearly integral, but $\mathbb C\otimes_{\mathbb R}\mathbb C\cong \mathbb C\times \mathbb C$ is not integral.
Regarding the second example, consider the field of rational functions $\mathbb F_p(t)$ for a prime number $p$ and the $\mathbb F_p(t)$-algebra $\mathbb F_p(t)[X]/(X^p-t)$. As the polynomial $X^p-t$ is irreducible, this is a field, hence a domain. However, tensoring with the algebraic closure of $\mathbb F_p(t)$ yields $\overline{\mathbb F_p(t)}[X]/(X^p-t)\cong \overline{\mathbb F_p(t)}[X]/(X-\xi)^p$ where $\xi$ is the (unique) $p$th root of $t$. Here the residue class of $X-\xi$ is a nonzero nilpotent element.
Edit: The equality $\overline{\mathbb F_p(t)}[X]/(X^p-t)\cong \overline{\mathbb F_p(t)}[X]/(X-\xi)^p$ holds since $(X^p-t)=(X-\xi)^p$ in $\overline{\mathbb F_p(t)}[X]$ holds. To see this, note that in any commutative ring of characteristic $p$, the "freshman's dream" $(a+b)^p=a^p+b^p$ holds for any elements $a,b$ in the ring. As $\overline{\mathbb F_p(t)}[X]$ is of characteristic $p$, we find $(X-\xi)^p=X^p-\xi^p=X^p-t$ by definition of $\xi$. Hence $X^p-t$ is completely inseparable, i.e. has a single root of multiplicity $p$.