I try to add a field in the schema.xml and can't make it work

You need to modify the solrconfig to return the field by default - in addition, though, it's likely that apachesolr (i.e., the Drupal module) won't see this because it requests a specific set of fields from the Solr server.

in apachesolr_search.module:

function apachesolr_search_basic_params($query) {
  $params = array(
    'fl' => 'id,nid,title,comment_count,type,created,changed,score,path,url,uid,name',
    'rows' => variable_get('apachesolr_rows', 10),
    'facet' => 'true',
    'facet.mincount' => 1,
    'facet.sort' => 'true'
  );
  return $params;
}

It's easy to add to this however, see, for example, the apachesolr_image.module hook implementation:

/**
 * Implementation of hook_apachesolr_modify_query().
 */
function apachesolr_image_apachesolr_modify_query(&$query, &$params) {
  // Also retrieve image thumbnail links.
  $params['fl'] .= ',ss_image_relative';
}