The Child Tax Credit can be taken only by the parent who has listed one or more eligible children as a dependent on the parent's tax return. The requirements for who is a dependent are listed on this IRS webpage: Dependents.
So, who does the child live with all year? Who pays for more than 50% of the child's support? How old is the child? The answers will determine who is able to claim the child.
Claiming children is not a negotiating game where adult pass children around back and forth to maximize tax refunds - when you apply the rule (see link above), normally only one adult will be able to claim a child - especially if there is a divorce decree document that specifies who can claim a child.
You are not required to have a dependent to be able to take the EITC; however, having one or more dependents increases the size of the credit.
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