If a divorce decree doesn't name who gets to claim a child as a dependent, then who gets to claim that child? What is both spouse try to claim that child? Who gets priority?

How come the decree failed to consider that important aspect.
If that is the case, move for clarification from the court that issued the decree. It is only the court who get to decide on that matter.
At most, if the child was still in tender years, the mother will held the child as her dependent.
If the child reached legal age or even if still a minor but can discern, the child can choose which parent he/she moves in.
In case of conflict, the court will consider the paramount welfare of the child.

for income tax purposes, i believe that the IRS states that the parent who files first gets to claim the dependency; assuming that both are qualified to claim the child as a dependent.

Jackie,

You are right.....partially.

This is what usually happens: Both parents may be entitled to claim a child as a dependent, but one spouse rushes to get his/her tax return filed first and claims the dependency exemption.

What is supposed to happen:

If more than one person files a return claiming the same qualifying child and only one of the persons is the child's parent, then the parent can claim the child as a dependent.

If two of the persons are parents of the child and they do not file a joint return together, then the parent with whom the child lived for the longer period of time during the year can claim that child as a dependent.

If two of the persons are parents of the child, they do not file a joint return together, and the child lived with each parent the same amount of time during the year, then the parent with the higher adjusted gross income (AGI) can claim the dependency exemption.

If one spouse was not entitled to claim the exemption but did anyway, then how does it get corrected? Is a protest filed?