Computational algebra: where? I'm on my last semester of a math B.Sc. and about to start studying for a math M.Sc in the same institute.
It now seems like a good time to start thinking of a PhD.
I'm interested in both algebra and algorithms. So I read a little in some computational algebra books (comp. group theory to be precise) and it looks like a great choice for me.
My questions:
1. Is this a "hot topic" in mathematics? Are there many serious people researching it?
2. Where are the major places in the world to research comp. algebra and to get a PhD?
Thank you!
 A: Some other suggestions:
The University of Auckland (Eamonn O'Brien and Marston Conder)
The University of Warwick (Derek Holt)
I did my MSc with Eamonn O'Brien and my PhD with Derek Holt, and can recommend them both as supervisors.  You are welcome to contact me by email (which you can find on my website) if you have any specific questions about either university.
A: The first piece of advice I would give you is not to prematurely limit yourself to a narrow area, unless your M.Sc only takes a year. If it takes at least two years, then you will still learn plenty of exciting mathematics before you have to make a reasonably definitive choice. If you are thinking of doing a PhD in the states, then even the PhD itself is so long, that being at a generally strong department might be more beneficial than to have an isolated area of expertise around you.
Having said that, if you are interested in computation mathematics of an algebraic nature (computational group theory, number theory, geometry, more general algebra), then a natural choice would be some place where one of the big computer algebra packages is being developed. The sites I list below also provide a good list of contributors, which will help you in choosing a shortlist of possible advisors. Some examples include

*

*MAGMA in Sydney

*GAP in various places; this one is particularly focused on group theory

*PARI/GP in Bordeaux

*SAGE in Washington, whose site also provides a nice map of where the developers are based. Thanks to Suvrit for fixing this unforgivable omission.

If I can think of more, I will add them here, but these should give you a good place to start. You can also browse through some of the dedicated journals, like the Journal of symbolic computation or the LMS Journal of computation and see who the prolific contributors are. Reading some of those articles might also help you decide whether this is what you want to do.
As for your question whether this is a "hot topic", that's harder to answer. It certainly is in the places mentioned above and in various others, where there are strong computational algebra groups, but I know of some places where this area of mathematics is frowned upon by the more theoretical community. But I suppose that that's probably true of almost any area of mathematics and you shouldn't let that deter you.
A: For computational algebra, I think there are some more places. You may want to look at
Symbolic Computation Group in Waterloo University (Canada).
Research Institute for Symbolic Computations in Linz (Austria)
Computer Algebra in Kaiserslautern (Germany), where Singular is (Wayback Machine)
Centre for Interdisciplinary Research in Computational Algebra (University of St Andrews, UK). (Wayback Machine)
CoCoA Sytem : Computations in Commutative Algebra.(Italy)
Macaulay2 : Computations in Algebraic Geometry and Commutative Algebra. (USA)
Moreover, I think in US, there are also some other places, like UIUC, North Carolina State Univ.  . .
I hope those information may be useful.
Good luck.
Best,
John
A: This might not directly answer your question, but there are a couple of books in the "Lecture Notes in Computer Science" series which are a collection of papers in Applicable/Applied algebra. I am including google books links for these. This might give you an idea of the kind of work taking place in these areas and the people involved. In recent times, I think, there have been a good number commutative algebraists, algebraic geometers, number theorists and combinatorialists working parallel in these areas.
Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: 10th International Symposium, AAECC-10, San Juan de Puerto Rico, Puerto Rico, May 10-14, 1993. Proceedings
Applicable Algebra, Error-Correcting Codes, Combinatorics and Computer Algebra: 4th International Conference, AAECC-4, Karlsruhe, FRG, September 23-26, 1986. Proceedings
