Irish players, officials and fans were angered by a clear handball from French captain Thierry Henry, who cushioned the ball to set up William Gallas to score the tie's winning goal in the first half of extra time in Paris.
The result booked France a spot at next year's World Cup in South Africa, breaking Irish hopes of reaching the finals for the first time since 2002.
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"The handball was recognised by the FIFA commissioner, the referee observer and the match officials, as well as by the player himself," read an FAI statement.
"There is precedent for the invalidation of such results. In 2005, the bureau of the FIFA World Cup organising committee reached a decision to invalidate the result of a World Cup qualification match between Uzbekistan and Bahrain on the basis of a 'technical error by the referee of the match'.
"The Football Association of Ireland is hoping that FIFA and its disciplinary committee will, on behalf of football fans worldwide, act in a similar fashion so that the standards of fair play and integrity can be protected."
Fifa have acknowledged receipt of the letter, and issued the following statement:
"As stated in Law 5 of the Laws of the Game, and also in Article 13, paragraph 6 of the Regulations of the 2010 FIFA World Cup, 'no protests may be made about the referee's decisions regarding facts connected with play. Such decisions are final, unless otherwise stipulated in the FIFA Disciplinary Code.'"
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- Football.co.uk staff