I have legal full custody but my ex-wife refuses to send my son back home, so he lived with her over half the year and she claimed him.

The tax code is based on the dependent physically residing with the parent for over one-half of the year for the parent to be able to claim the child as a dependent.

You have a legal issue that has to adjudicated by a court.

To be a Qualifying Child -

1. The child must be your son, daughter, stepchild, foster child, brother, sister, half brother, half sister, stepbrother, stepsister, or a descendant of any of them.
2. The child must be (a) under age 19 at the end of the year, (b) under age 24 at the end of the year and a full-time student or (c) any age and permanently and totally disabled.
3. The child must have lived with you for more than half of the year. Temporary absences while away at college are considered living with you.
4. The child must not have provided more than half of his or her own support for the year.
5. If the child meets the rules to be a qualifying child of more than one person, you must be the person entitled to claim the child as a qualifying child.
6. The child must be a U.S. citizen or U.S., Canada or Mexico resident for some portion of the year.
7. The child must be younger than you unless disabled.

Legal custody and tax custody are totally different.     

The IRS only recognizes physical custody (which parent the child lived with the greater part, but over half, of the tax year. That parent is the custodial parent; the other parent is the noncustodial parent.)

Who can claim the exemption and credits depends on who is the custodial parent. (By the IRS definition of custodial parent for tax purposes - this is not the same as the legal custody that a court might grant.).

The test that the IRS uses to determine the custodial parent is where the child lived for more than 1/2 (or greater part) of the year. The IRS will go so far as to require counting the nights spend in each household - that person is the custodial parent for tax purposes (if exactly equal and more than 183 days - The custodial parent is the parent with the highest AGI, if less than 183 days then neither parent has custody so the child cannot be claimed by either parent). And yes they are that picky.

See Custodial parent and noncustodial parent in Pub 501

https://www.irs.gov/publications/p501#en_US_2018_publink1000220906

Only the Custodial parent can claim: (Child would be listed as non-dependent EIC & CC only)
-Head of Household
-The Earned Income Credit
-The Child and Dependent Care Credit
-The Health Coverage Tax Credit

The non custodial parent can only claim: (Child would be listed as dependent)
- The child as a dependent
- The Child Tax Credit or credit for other dependents

But only if specifically specified in a pre-2009 divorce decree, separation agreement or the custodial spouse releases the exemption with a signed 8332 form - after 2009 the IRS only accepts a signed 8332 form that must be attached to the non-custodial parents tax return.

This does NOT mean that the custodial parent can ignore any Decree or court order allowing the non-custodial parent to claim the exemption - they can be required to issue the 8332 form. They could be required by the court to do so or be in contempt.

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Post-1984 and pre-2009 divorce decree or separation agreement. If the divorce decree or separation agreement went into effect after 1984 and before 2009, the noncustodial parent may be able to attach certain pages from the decree or agreement instead of Form 8332. The decree or agreement must state all three of the following.

1. The noncustodial parent can claim the child as a dependent without regard to any condition, such as payment of support.
2. The custodial parent will not claim the child as a dependent for the year.
3.The years for which the noncustodial parent, rather than the custodial parent, can claim the child as a dependent.

The noncustodial parent must attach all of the following pages of the decree or agreement to his or her tax return

* The cover page (write the other parent's social security number on this page).
* The pages that include all of the information identified in items (1) through (3) above.
* The signature page with the other parent's signature and the date of the agreement.

Post-2008 divorce decree or separation agreement. The noncustodial parent cannot attach pages from the decree or agreement instead of Form 8332 if the decree or agreement went into effect after 2008. The custodial parent must sign either Form 8332 or a similar statement whose only purpose is to release the custodial parent's claim to an exemption for a child, and the noncustodial parent must attach a copy to his or her return. The form or statement must release the custodial parent's claim to the child without any conditions. For example, the release must not depend on the noncustodial parent paying support.