Answer
Yes, the API can be downloaded and installed locally; however, using the hosted ArcGIS JavaScript API is recommended. To use the hosted version, add the following script tag to the page:
<script type="text/javascript" src="http://serverapi.arcgisonline.com/jsapi/arcgis/?v=2.0"></script>
For more information, please see the following help topic for the ArcGIS API for JavaScript.