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.


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.