Yes phpMyAdmin works okay with root user.
I have tried: $link = mysql_connect("localhost", "root",""

or die("Kunne ikke forbinde : " . mysql_error());
and
$link = mysql_connect("localhost", "root"

or die("Kunne ikke forbinde : " . mysql_error());
same result: Warning: mysql_connect() [function.mysql-connect]: [2002] Et forsøg på at oprette forbindelse mislykkedes, fordi den part, (trying to connect via tcp://localhost:3306) in ....
here below I have copied users:
Bruger Vært Kodeord Globale privilegier 1 Tildel Handling
korthaarklubben localhost Ja ALL PRIVILEGES Ja
root localhost Nej ALL PRIVILEGES Ja
Afmærk alt / Fjern alle mærker
As you maybe have noticed I am from Denmark, and som of the text are on danish, I hope you understand it anyway :-)