4
$\begingroup$

I am trying to compute the relative Brauer group of the cyclic Galois extension $L=\mathbb Q[x]/(x^3-3x+1)$ of $\mathbb Q$. I know that $$ \mathrm{Br}(L/\mathbb Q)\cong H^2(G,L^*)\cong\mathbb Q^*/N(L^*) $$ where $G=\langle\sigma\rangle$ is the Galois group and $N$ the field norm, $N(l)=l\cdot\sigma(l)\cdot\sigma^2(l)$.

According to Michael Artin's book Algebra, the roots of the polynomial are $\eta_1=\zeta+\zeta^8$, $\eta_2=\eta_1^2-2=\zeta^2+\zeta^7$ and $\eta_3=-\eta_1-\eta_2=\zeta^4+\zeta^5$ where $\zeta=\exp(2\pi i/9)$.

Writing an arbitrary element $l\in L$ as $a+b\eta_1+c\eta_2$ with $a,b,c\in\mathbb Q$, I have computed that $$ N(l)=a^3-b^3-c^3-3ab^2-3ac^2+3abc+6b^2c+6bc^2. $$ Can you help me compute the quotient $\mathbb Q^*/N(L^*)$?

I know that $\mathbb Q^*\cong\{\pm1\}\times\prod_p\mathbb Z$ but am having trouble determining the image of $N$ because of the additive operations appearing in the explicit expression above.

$\endgroup$
  • $\begingroup$ I don't think this is possible to compute explicitly $N(L^*)$ (I don't have any formal argument, it's just usually very hard to compute norm groups). My opinion is that you should rather try to use Brauer-Hasse-Noether. $\endgroup$ – Captain Lama Apr 23 '16 at 8:44
  • 1
    $\begingroup$ Here are some thoughts, I havent had time to do any calculations. First is (the ring of integers of) $L$ a PID, that is does class number $=1$. If so your life is very easy, just find the factorisations of the primes. If not I would start by compairing your object with the product of the localizations (see Brauer-Hasse-Noether) above. Consider the far easier problem of $\mathrm{Br}(\mathbb Q(i)/\mathbb Q)$. It is an infinite product of $\mathbb{Z}_2$, one for each prime $\equiv 3 (\mod 4)$. $\endgroup$ – Rene Schipperus Apr 23 '16 at 13:51
3
$\begingroup$

I agree with Captain Lama, it would be hardly possible to determine the relative Brauer group $Br(L/\mathbf Q)$ just by norm computation. I sketch here a theoretical approach.

By definition, $Br(L/\mathbf Q)$ is the kernel of the natural map $Br(\mathbf Q) \to Br(L)$ (which is the restriction map, cohomologically speaking). For any number field $K$, it is known by cohomological CFT that $Br(K)$ can be identified canonically with the kernel of the « sum of coordinates » map inside the direct sum of all the local Brauer groups $Br(K_v)$, $v$ running over all the primes of $K$.

Thus we are brought back to the determination of these $Br(K_v)$, and their fuctorial behaviour w.r.t. the restriction map. But the answers are known by local CFT : for a finite $v$, $Br(K_v) \cong \mathbf Q/ \mathbf Z$, and in an extension $L_w/K_v$ of degree $n_v$, the restriction $Br(K_v) \to Br(L_w)$ corresponds to multiplication by $n_v$ in $\mathbf Q/\mathbf Z$ . See e.g. Serre’s « Local Fields », chapters 10 and 13. Thus $Br(L_w/K_v) \cong \mathbf Z/n_v$.

Coming back to your problem, it only remains to compute the degrees of the local extensions $L_w/\mathbf Q_p$ associated to the polynomial $X^3 - 3X + 1 $. Note that the method still works when replacing $\mathbf Q$ by any $K$.

$\endgroup$

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.