Odeslání formuláře bez refreshe

<script>
            Casovac();

            function NewHodnota() {
                hodnota = document.getElementById('hodnota');
                if (window.XMLHttpRequest) xmlhttp = new XMLHttpRequest();
                else xmlhttp=new ActiveXObject("Microsoft.XMLHTTP"); 
                xmlhttp.onreadystatechange=function() {
                  if (xmlhttp.readyState==4 && xmlhttp.status==200) {
                        hodnota.innerHTML = xmlhttp.responseText;
                }    
              }    
                xmlhttp.open("GET", "./../../?page=chat_list", true);
                xmlhttp.send(); 
                Casovac();
            }
    
            function Casovac() { 
                 setTimeout("NewHodnota()", 1000);
            }
</script>
<span id='hodnota'></span>

<!--http://www.9lessons.info/2009/04/submit-form-jquery-and-ajax.html-->    

<script type="text/javascript" src="http://ajax.googleapis.com/ajax/libs/jquery/1.3.0/jquery.min.js"></script>
<script type="text/javascript" >
$(function() {
$(".subChat").click(function() {

var textChat = $("#textChat").val();
var dataString = 'text_chat='+ textChat;

$.ajax({
type: "POST",
url: "script.php",
data: dataString
});

return false;
});
});
</script>

<form method="post">
<input id="textChat" name="text_chat" type="text" value=""/>
<input type="submit" name="submitname" value="Odeslat" class="subChat"/>
</form>

<?php
if(isset($_POST['text_chat']))
{
  
    $heslo = "vertrigo";        
    $server = "localhost";
    $login = "root";
    $database = "system";

    MySQL_Connect($server, $login, $heslo); 
  MySQL_Select_DB($database);
  
  $name=$_POST['text_chat'];
  
    mysql_query("INSERT INTO zkouska (id, text) VALUES ('', '".$name."')");
  
}