Image of the trace map of ring of integers

Question

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.

Answer 1

Question 1. Yes, and this follows from the results of Chapter VIII in Weil: Basic Number Theory. See especially Corollary 2 of Proposition 4 in that chapter.

Question 2. In general, the exponent of $p$ in $d$ equals $\lceil g_p/e_p \rceil$, where $g_p$ is the exponent of any prime $\mathfrak{p}\mid p$ in the different of $L_\mathfrak{P}/\mathbb{Q}_p$, and $e_p$ is the ramification degree at $p$. The exponent $g_p$ itself can be calculated in terms of higher ramification degrees of $L_\mathfrak{P}/\mathbb{Q}_p$. These facts can be found in the same chapter that I quoted above: see especially Proposition 4 and display (10) in that chapter.