I am looking for an introductory text to the subject of Cyclic Algebras, and in particular ones defined over a local field.
A cyclic Algebra, to the best of by understanding is defined as follows: Let $F$ be a local field, and let $E/F$ be a finite cyclic extension of degree $n$ with $G=\langle \sigma\rangle=Gal(E/F)$ be the Galois group. Let $\alpha\in F$. A cyclic $F$-algebra $A$ is defined w.r.t $(E,\sigma,\alpha)$ as follows: Let $R$ be the twisted polynomial $F$-algebra $E[T]_\sigma$, where the elements are polyminals $\sum_{i=0}^{n-1}a_{i}T^i$ and with multiplication defined by the rule $l\cdot T=T\cdot\sigma(l),\:l\in E$.
Then $A$ is given as $A:=R/\langle T^n-\alpha\rangle$ (please let me know if there is anything wrong in my definition).
Of main concern for me would be:
- Structure theorems for cyclic algebras, and in particular for the cases where $F$ is a local field and $E$ an unramified extension
- Any investigation into the representation theory of the groups $A^\times$ (the multiplicative group of $A$), and $SL_1(A)$ (the group of elements in $A$ of reduced norm $1$).
I would very much appreciate any reference that could be offered
Thank you :-)