Add extra fields to Add New Category page

I need to add an extra field to the Add New Category page. Currently it only has fields for Name, Slug, and Description. Essentially, I need to have a second description text box.

I use the current Description field at the bottom of each category page, but now I need to have additional information specific to the category (and different from the bottom) displayed at the top.

How can I add this additional field?

Below is what I am trying to achieve:

If you want a plugin solution, advanced custom fields can do this.