Did Hector Kill Ajax in the Trojan War

In the Warner Bros. movie Troy, Ajax sees Achilles dashing madly ahead of the rest of the Greeks to land on the shores of Troy. This impels him to push his own men harder, to row faster. A big man, Ajax does his share of hacking away at the enemy until Hector kills him. The legend of Ajax is different. Ajax was the son of the king of Salamis, Telamon. As a suitor of Helen, Ajax had been obligated by the oath of Tyndareus to fight for the return of Helen to Menelaus. He fulfilled his obligation by leading twelve ships with troops from Salamis to Troy. There he fought in single combat against Hector, but he was not slain. Instead, Hector and Ajax exchanged gifts. Hector gave Ajax his sword and Ajax gave Hector his belt. It was by this belt that Achilles dragged Hector through the dirt. After Achilles was killed, his armor was awarded to the Greek thought most heroic. Ajax thought that with Achilles dead, the prize should have been his. Instead, the prize was awarded to Odysseus. Ajax went mad and tried to kill Odysseus and other Greeks. Athena intervened and made him see Greeks where there were really cattle. When Ajax recovered, he was mortified by his deeds, although still aggrieved by the slight, and so killed himself using the sword Hector had given him.