Let $F=\mathbb Q(\zeta_5,\sqrt[5]2)$ and $K=\mathbb Q$ where $\zeta_5$ is a primitive $5$th root of unity and let $p=73$ be a prime in $K$. Fix primes $\mathfrak p$ and $\mathfrak q$ above $73$ in $\mathbb Q(\zeta_5)$ and $F$ respectively.

I know that no ramification occurs in $\mathbb Q(\sqrt[5]2)/\mathbb Q$ since $X^5-2$ has distinct roots mod $73$ but would this imply that $\mathfrak q$ is unramified over $\mathfrak p$?

Many thanks for your help.


Yes, the only ramification in sight here is at $2$ and $5$. One can conclude that a prime foreign to these two is unramified in $F$ without looking at the factorization of $X^5-2$. Now, you notice that the first power of $73$ that is $\equiv1\pmod5$ is the fourth, so that the residue field of any prime of $\Bbb Q(\zeta_5)$ above $73$ must have $73^4$ elements. In other words, the residue-field extension degree (“$f$”) is $4$, which means that there’s only one prime of $\Bbb Q(\zeta_5)$ above $73$. The polynomial $X^5-2$ factors completely into linears over that big residue field, and you conclude that there are five primes of $F$ above $73$.

  • $\begingroup$ By the big residue field do you mean $\mathcal O_F/\mathfrak q$? Would I also be right to argue like this: if $\mathfrak q/ \mathfrak p$ is ramified then since $[F:\mathbb Q(\zeta_5)]=5$, by the multiplicativity of $e$, $e_{\mathfrak q/73}$ is divisible by $5$. But since $X^5-2$ has distinct roots mod $73$, there is no ramification in $\mathbb Q(\sqrt[5]2)/\mathbb Q$ and so this is not possible as $[F:\mathbb Q(\sqrt[5]2)]=4$. Apologies for writing it out like this- I am new to the subject and I just want to get the details right. Regards $\endgroup$ – Gwan May 15 '15 at 16:21
  • 1
    $\begingroup$ Yes, that’s what I mean by “big residue field”. If $\mathfrak q/\mathfrak p$ were ramified, then one of the primes of $F$ over $\mathfrak p$ would have ramification greater than $1$. Of course, in this case, the extension is normal, so that all the primes would behave similarly, so the only possibilities are one prime with either $e=5$ or $f=5$, or five primes with $e=f=1$. But I insist: if you adjoin the fifth roots of unity, the only ramification is above $5$; and if you adjoin the fifth roots of something else, the only ramification is above $5$ & the divisors of that something else. $\endgroup$ – Lubin May 15 '15 at 17:28
  • $\begingroup$ (Sorry, ran out of characters.) As a result, ramification above $73$ was out of the question from the outset. $\endgroup$ – Lubin May 15 '15 at 17:30

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.