Ajax does not fire my function

In my javascript file:

jQuery(document).ready( function($) {
        data={'action': 'check_email_exists','umail': field.value};
        jQuery.post( ajax_url, data, function(response) {
          console.log(data);
          console.log(response);
          if(response=='true'){
            warning=document.getElementById('email_r');
            warning.innerHTML='This email is already taken';
            warning.style.display='inline';
            warning.scrollIntoView();
            return true;
          }
        });
      });

In my plugin’s php file:

add_action('wp_ajax_nopriv_checkemail', 'check_email_exists');
add_action( 'wp_enqueue_scripts', 'call_external_scripts' );

function check_email_exists(){
  global $wpdb;
  $sql="SELECT id FROM ".$wpdb->prefix."users WHERE user_email = '".$_POST['umail']."';";
  //echo $sql;
  $results = $wpdb->get_results($sql);
  if($results){
    echo 'true';
  }else{
    echo 'false';
  }
  exit;
}

The function is not called.

If I add this to the php file:

if($_POST['action']){
    global $wpdb;
    if($_POST['action']=='check_email_exists'){
      $sql="SELECT id FROM ".$wpdb->prefix."users WHERE user_email = '".$_POST['umail']."';";
      //echo $sql;
      $results = $wpdb->get_results($sql);
      if($results){
        echo 'true';
      }else{
        echo 'false';
      }
    }//*/
    exit;
  }//*/

Then it works. The javascript gets ‘true’ if the user exists and ‘false’ if he doesn’t.

The problem is that the site is borked. Any act of saving a page or a post or editing a user results in a whitescreen.

Baffling…