I am trying to study cyclic algebras $(\chi,a)$ over a local field $K$ (specifically of characteristic 0). Given a cyclic Galois extension $L/K$ with isomorphism $\chi\colon Gal(L/K)\rightarrow\mathbb{Z}/n\mathbb{Z}$ with $\chi(\sigma)=\bar{1}$, and an element $a\in K^*$, the cyclic algebra $(\chi,a)$ is defined to be the central simple algebra $$L[y]/(y^n=a,\ y^{-1}xy=\sigma(x)).$$ It would be nice to always be able to reduce to the case where the extension $L/K$ is unramified, since there is a unique unramified extension $K_n/K$ of degree $n$. So far, I have found that any cyclic algebra as above is equal to a cyclic algebra $(\varphi,b)$, where $\varphi\colon Gal(K_n/K)\rightarrow\mathbb{Z}/n\mathbb{Z}$ is an isomorphism and $b\in K^*$ (see Theorem 8 in these notes).
I am particularly interested in finding out which cyclic algebras can be reached if we restrict the choice of $a$ to a certain subset of $K$. So in order to use the reduction step above, I have to know what this element $b$ can be. Can we always take it to be $a$ so that $(\chi,a)=(\varphi,a)$? I have tried to solve this by looking at the relevant chapter in Gille and Szamuely's book on central simple algebras (in particular Proposition 2.5.3 and Lemma 2.5.4), but I don't see a reason why $a$ would be equal to $b$.
If it turns out to be impossible to control $b$, another possible reduction step would be to note that the extension $L/K$ can be split into an unramified part and a totally ramified part. I haven't looked into this in detail, but maybe the element $[(\chi,a)]$ in the Brauer group $Br(K)$ can then be factored into $[(\varphi,a)][(\psi,a)]$, where $\varphi$ and $\psi$ correspond to the unramified and totally ramified extensions. This way I might still be able to determine $[(\chi,a)]$ by only considering the unramified case and the totally ramified cases.
To sum up, I have the following two questions:
- When reducing $(\chi,a)$ to the unramified version $(\varphi,b)$, is it possible to write $b$ in terms of $a$?
- If not, can we somehow factor $[(\chi,a)]$ so that we only have to consider the cases where $L/K$ is either unramified or totally ramified? (while still keeping the same element $a$)