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. 



Image of the trace map of ring of integers
**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.


