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 :-)

  • 3
    The book Associative Algebras by Pierce should be useful. – Mariano Suárez-Álvarez 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 3 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.

Your Answer

 

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.