Let $L/\mathbb{Q}$ be a finite Galois extension, and let $\mathcal{O}_L$ be the ring of integers of $L$.

We have $tr_{L/\mathbb{Q}}(\mathcal{O}_L)=d\mathbb{Z}$ for some $d\geq 1.$

**Fact.** $d=1$ if and only if $L/\mathbb{Q}$ is tamely ramified.

**Question 1.** Let $p$ be a prime number. Is it true that $tr_{L/\mathbb{Q}}(x)=0 \pmod p$ for all $x\in\mathcal{O}_L$ if and only if $p$ is wildly ramified ?

**Question 2.** Is the value of $d$ computable in terms of the arithmetic of $L/\mathbb{Q}$ ?

For Question 1, it would suffice to prove the following fact: if $\mathfrak{p}$ lies above $p$ and $e\geq 2$, then the trace map of the $\mathbb{F}_p$-algebra $\mathcal{O}_L/\mathfrak{p}^e$ is zero if and only if $p\mid e$. I almost convinced myself it is true, but I can't get to a complete proof...

Any help/reference woud be greatly appreciated.