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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 :-)

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The book Associative Algebras by Pierce should be useful. – Mariano Suárez-Alvarez Mar 5 '13 at 3:58
Thanks, Pierce is my main reference- I was looking for an extra reference :) – kneidell Mar 5 '13 at 9:57
up vote 2 down vote accepted

For $L/k$ is a cyclic extension of degree $n$ with Galois group $\mathrm Gal(L/k)=:G$ with a generator $\sigma$ and let $a\in K^*$. The ring $(a,L/k,\sigma):=\oplus_{i=0}^{i=n-1} Le^i$ with the multiplication $e^n=a$ and $e\alpha=\sigma(\alpha)e$ with $\alpha\in L$ is called cyclic algebra.It can be proved that for a local field $k$ such a cyclic algebra $A$, $\mathrm ind~A=\mathrm exp~A$. Moreover, we can get an isomorphism $\mathrm inv_k:\mathrm Br(k)\rightarrow \mathbb Q/\mathbb Z$.

You may see Serre's Local field; Chapter on Local Class field theory. Here he defines map $\mathrm inv_k$. Also you may find a book titled 'Central Simple Algebra and Galos Cohomology' by Philippe Gille and Tamas Szamuely useful.

$\textbf{Edit:}$ For $SL_1(A)$ and its relation to Whitehead grou $SK_1(A)$ you may look at the Ch. 6, section 18.3-18.4 of the book titled 'Algebraic groups and their birational invariants' by Voskresenskii.

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