How do I find the authoritative name-server for a domain name?

You'll want the SOA (Start of Authority) record for a given domain name, and this is how you accomplish it using the universally available nslookup command line tool:

command line> nslookup
> set querytype=soa
> stackoverflow.com
Server:         217.30.180.230
Address:        217.30.180.230#53

Non-authoritative answer:
stackoverflow.com
        origin = ns51.domaincontrol.com # ("primary name server" on Windows)
        mail addr = dns.jomax.net       # ("responsible mail addr" on Windows)
        serial = 2008041300
        refresh = 28800
        retry = 7200
        expire = 604800
        minimum = 86400
Authoritative answers can be found from:
stackoverflow.com       nameserver = ns52.domaincontrol.com.
stackoverflow.com       nameserver = ns51.domaincontrol.com.

The origin (or primary name server on Windows) line tells you that ns51.domaincontrol is the main name server for stackoverflow.com.

At the end of output all authoritative servers, including backup servers for the given domain, are listed.

You used the singular in your question but there are typically several authoritative name servers, the RFC 1034 recommends at least two.

Unless you mean "primary name server" and not "authoritative name server". The secondary name servers are authoritative.

To find out the name servers of a domain on Unix:

  % dig +short NS stackoverflow.com
 ns52.domaincontrol.com.
 ns51.domaincontrol.com.

To find out the server listed as primary (the notion of "primary" is quite fuzzy these days and typically has no good answer):

% dig +short  SOA stackoverflow.com | cut -d' ' -f1
ns51.domaincontrol.com.

To check discrepencies between name servers, my preference goes to the old check_soa tool, described in Liu & Albitz "DNS & BIND" book (O'Reilly editor). The source code is available in http://examples.oreilly.com/dns5/

% check_soa stackoverflow.com
ns51.domaincontrol.com has serial number 2008041300
ns52.domaincontrol.com has serial number 2008041300

Here, the two authoritative name servers have the same serial number. Good.

On *nix:

$ dig -t ns <domain name>