You only need to connect to the database once. Your code will attempt to reconnect each time through the loop

my $dbh = DBI->connect("DBI:mysql:database=proyecto;host=localhost", "
+root", "xyz123", {'RaiseError' => 1});

# Pre-prepare using placeholders
my $sth_a = $dbh->prepare("SELECT dept, course FROM final WHERE pat=?"
+);

my $sth = $dbh->prepare("SELECT DISTINCT PAT FROM patentes");  $sth->e
+xecute();
while (my ($pat) = $sth->fetchrow_array()) {  # You don't need an arra
+y here
    $sth_a->execute($pat);
    while (my ($dept,$course) = $sth_a->fetchrow_array()) { 
        print "$pat, $dept, $course\n";

        # Here you can create and populate your new tables as needed 

    }
}
$sth->finish()
$sth_a-finish();
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Update
Don't forget to check whether or not your new table exists in that inner loop. You don't want to repeatedly re-create the tables.