Does making a field unique make it indexed? If I make a unique constraint on a field, do I also need to make an index on that field in order to get a scalable insert time? Or is this done for me (even if the index it uses isn't publicly accessible?)

Specifically, I'm working with Apache Derby for prototyping, although I will probably be moving it to MySQL in the semi-near future. I'm also hoping there might be something in the SQL standard that says something about this.

I will never have a need to search by this field, so I would rather not make a useless index. But I'd rather have a useless index than have an O(n) insert time.
--EDIT--

My original answer (below) is probably not useful to you at all because it does not address the question of unique constraints. As others have said, these constraints are usually implemented with an implied unique index. In special cases this might not be true (eg disable novalidate for Oracle). 

The question could be: Is it possible to enforce uniqueness without an index? Generally speaking the answer is no though in some cases a Clustered Index will mean that the index and the table are the same object.

--END EDIT--

You said "I'd rather have a useless index than have an O(n) insert time.", but in general databases do not have O(n) insert time. There are two cases to consider:


A normal table with or without indexes: 

New rows are dumped at the top of the heap. The RDBMS probably only looks at 1 block, so not just O(1) but very small O(1).

If the table has indexes, a pointer to the row will be added to each. This will usually be an O(log(n)) operation.
A table with some sort of clustering going on, eg an Index Organized Table or cluster for Oracle, or a Clustered Index for SQL Server and others:

New rows are inserted into a particular block, which may cause the block to split or overflow, but whatever happens it is still O(log(n)) or better, caused by the b-tree or similar structure used to find the block.