Inspired by this:
The Rambam in Hilchos Melachim 6:1 writes:
אין עושין מלחמה עם אדם בעולם, עד שקוראין לו לשלום--אחד מלחמת הרשות, ואחד מלחמת מצוה
War is not waged with anyone in the world until they were offered peace...
and in 6:6 the Rambam includes Amalek in this rule:
אבל שבעה עממין ועמלק שלא השלימו, אין מניחין מהם נשמה
If the Seven Nations or Amalek do not accept peace, wipe them out.
In the story of King Saul's war on Amalek (Samuel 1 15:1-9), there is no mention of any offering of peace before Amalek is destroyed. Is there any reference to such a communication, and if not how does the Rambam know to include Amalek in this rule? I would have considered that as Hashem is in a constant state of war with Amalek (Shemos 17:16) and we are instructed to erase their memory (Devarim 25:19), there would be an objective desire to have the war with them, and making peace would not suffice.