Corporal Alexander was a member of Headquarters Battery, 4th Battalion, 11th Marines, 1st Marine Division. He was listed as Missing in Action while fighting the enemy at the Chosin Reservoir in North Korea on 1 December 1950. He was presumed dead on 30 December 1953. His remains were not recovered. His name is inscribed on the Courts of the Missing at the Honolulu Memorial.

Alexander was awarded the Purple Heart, the Combat Action Ribbon, the Korean Service Medal, the United Nations Service Medal, the National Defense Service Medal, the Korean Presidential Unit Citation and the Republic of Korea War Service Medal.