I'm not sure if the DBI primary_key methods are implemented in DBD::mysql. There is a MySQL-specific way to find the keys using $sth->{mysql_is_pri_key} like so:

#!perl -w
use strict;
use DBI;
my($db,$table)=('test','tmp1');
my $dbh=DBI->connect("dbi:mysql(RaiseError=1):dbname=$db");
$dbh->do(qq/DROP TABLE IF EXISTS $table/);
$dbh->do(qq/CREATE TABLE $table (id INTEGER PRIMARY KEY, name VARCHAR(
+10))/);
for my $field(qw( id name )) {
    my $sth = $dbh->prepare("SELECT $field FROM $table WHERE 1=0");
    $sth->execute;
    for my $k($sth->{'mysql_is_pri_key'}) {
        my $n = 'IS';
        $n .= " NOT" unless $k->[0];
        print "$field $n A PRIMARY KEY\n";
    }
    $sth->finish;
}
$dbh->do(qq/DROP TABLE $table/);
$dbh->disconnect;
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