Ajax was the son of King Telamon and Periboea, and the half brother of Teucer. He is portrayed as a towering figure, and a warrior of great courage.
In the Iliad, Ajax is notable for his abundant strength and courage, and was chosen by lot to meet Hector in a duel which lasts most of a whole day. The encounter ended in a draw, with Ajax presenting Hector with a purple sash and Hector giving Ajax his silver sword. The second fight between the two heros occurs when the Trojans break into the Mycenean camp, and the fight takes place amongst the Greek ships. Ajax is responsible for the death of many Trojan lords, including Phorcys. According to Hyginus, in total, Ajax killed 28 people at Troy.

Traditionally, the Trojan War arose from a sequence of events beginning with a quarrel between the goddesses Hera, Athena and Aphrodite. Eris the goddess of discord, was not invited to the wedding of Peleus and Thetis, and so arrived bearing a gift. A golden apple, inscribed “for the fairest”.
Each of the goddesses claimed to be the “fairest”, and the rightful owner of the apple. They submitted the judgement to a shepherd they encountered tending his flock. Each of the goddesses promised the young man a boon in return for his favour. Power, wisdom, or love. The youth, in fact Paris, a Trojan prince who had been raised in the countryside, chose love, and awarded the apple to Aphrodite.
As his reward, Aphrodite caused Helen, the Queen of Sparta, and the most beautiful of all women, to fall in love with Paris. The judgement of Paris earned him the ire of both Hera and Athena, and when Helen left her husband, Menelaus, the Spartan king, for Paris of Troy, Menelaus called upon all the kings and princes of Greece to wage war upon Troy.
Menelaus’ brother Agamemnon King of Mycenae, led an expedition of Achaean troops to Troy and besieged the city for ten years because of Paris’ insult. After the death of many heroes, including the Achaeans, Achilles, Ajax and the Trojans Hector and Paris, the city fell to the ruse of the Trojan Horse.
The Achaeans slaughtered the Trojans, except for some of the women and children whom they kept or sold as slaves. They desecrated the temples, thus earning the wrath of the gods. Few of the Achaeans returned safely to their homes, and many founded colonies in distant shores. The Romans later traced their origin to Aeneas, Aphrodite’s son and one of the Trojans, who was said to have led the surviving Trojans to modern day Italy.