Can a non-custodial parent file for earned income credit?

Is it legally allowed for the non-custodial parent to take un-earned income credit for their child?
-Please give websites if possible.

the EARNED Income Credit is only available to the parent (or guardian in some cases) whom the child lived with for more than half the year. This is true even in the case where the non-custodial parent claims the exemption and Child Tax Credit (where applicable) for that child. In other words, the non-custodial parent can claim the exemption and the Child Tax Credit, and the custodial parent can claim the Head of Household filing status, the Earned Income Credit, and the Child and Dependent Care Credit. All of these have other requirements before they can be claimed on a return, but that is the way the IRS breaks it down.
On a side note - the non-custodial parent must have either a signed form 8332 (Release to Claim Exemption), a divorce decree, or other court document allowing him/her to claim the child's exemption - it is NOT automatic.
Hope this helps!

The noncustodial parent cannot claim the earned income credit on the basis of that child because the child did not live with that parent for more than half of the tax year, and therefore does not meet the residency test. The custodial parent may be able to claim the earned income credit.

No. See irs.gov - one place would be in the section of Publication 17 about EIC.

Only the custodial parent can claim the child for EIC, and they can do that even if the other parent claims the child as a dependent.