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.

  • $\begingroup$ 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
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    $\begingroup$ 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

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 $K$.


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