Yes, it does impact what credits can be claimed. The American Opportunity Tax Credit also has a refundable portion to it. If your child can be claimed, then they are not eligible for the refundable portion, they are only eligible for the portion that will decrease their tax due. If their income is less than the standard deduction or they have no tax due, they will not benefit from the credit at all.
You as the parent, if you are otherwise eligible to claim the credit, may be able to take both the refundable and non refundable portion of the credit.
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