Hi Geoff, you are perfectly right, there is no reason to expect the two characters to be irreducible.

@Francois, I did not see much examples in this book outside admissible representations. After all, their goal is to prove local Langlands correspondence. I would like to have a reference where one can get a feeling of the various types of representations: for example unitary versus non-unitary, continuous versus not strongly continuous, smooth but not admissible. For example if you look at $GL_2(R)$ with the discrete topology, "how many more" representations do you get from looking only at smooth ones. Basically, I want to see various ways of organizing representations of top. groups.

Thanks Joel for the nice example. Is there a good (recent) reference on representations of topological groups (locally compact is fine with me) which give a good overview of the various "categories" of representations (smooth, admissible etc....)?

I see, so I had obvious examples just under my nose! Thanks Paul

@grp, you still need to correct slightly your statement, if you take an irreducible cubic polynomial over $Q$ with 3 real roots then the splitting field does not include a root of unity of odd order. I think that what you meant was if $K$ is the splitting field of your polynomial over $F$ and if you embed $K$ in a radical extension, say $M$, then $M$ has to contain a root of unity of odd order. In fact the proof of this statement is straight forward from what I wrote in my second paragraph after Will's answer.

I just found this 4 pages document from Brian Conrad which discusses aspects related to my question: see math.stanford.edu/~conrad/210BPage/handouts/radreal.pdf

Hi @grp, that is quite a nice result. So in particuar using your observation we see that $z^n-1$ is positive solvable iff $\varphi(n)$ is a power of $2$ so iff $n=2^rq_1q_2\ldots q_s$ where the $q_i$'s are distinct Fermat's primes.

Of course one has to be careful by the meaning of a "reduced writting". For example $\sqrt[5]{1}$ would not be a reduced writting for me since the 4 complex 5-th roots of unity can be written in terms of $\sqrt[2]$. I'm not sure at this point that I would be able to define properly what I mean by a reduced writting... But definitely an expression like $\sqrt[n]{1}$ (for $n>1$) would not be reduced but its reduced writing would only involves a succession of $\sqrt[p]{}$ with $p$ prime dividing $\varphi(n)$.

There is at least the following easy observation: if $f(z)\in\mathbf{R}[z]$ has all its roots real then a (reduced) writting of a root of $f(z)$ in terms of radicals should only involve $\sqrt[m]{}$ with integers $m$ such that $\varphi(m)=2^r$, since for $p>2$ prime the group of $p$-th roots of unity cannot be embedded in the splitting field of $f(z)$ which is included in $\mathbf{R}$ by hypothesis.

I meant $a$ is a $p$-th power in $K$.

Note also that the proof that the $3$ real roots cannot be written in terms of radicals of positive quantities follows from the key observation that for a prime number $p$ and an arbitrary field $K$, the polynomial $z^p-a$ with $a\in K$ is either irreducible or $a$ is a $p$-th in $K$. Therefore at the last step of the sequence of fields appearing in the tower one must take a cube root. Since the splitting field is normal and real this would mean that the all 3rd roots of unity are real which is a contradiction.