What is the ruling on the salutation (salâm) given when finishing the prayer?

According to the preferred view in the Hanafi school, it is obligatory (wâjib) to give salâm at the end of the prayer, first turning the face to the right, then to the left. Deliberately omitting this requires the prayer to be repeated. If omitted unintentionally, nothing is required. In this salâm, saying the phrase “as-salāmu ʿalaykum wa raḥmatullāh,” the part “as-salām” is obligatory, while adding “ʿalaykum wa raḥmatullāh” is sunnah. According to another view, giving salâm to the right is obligatory, and to the left is sunnah. According to all imams, exiting the prayer is accomplished with a single salâm, and with this, the prayer ends. (Ibn al-Humām, Fath al-Qadīr, 1/320; Zaylaʿī, Tabyīn, 1/125-126) According to the Shafiʿis, who consider the first salâm to be obligatory (farḍ), omitting it invalidates the prayer. (Shirbīnī, Mughnī al-Muḥtāj, 1/385)

Source: T.C. Diyanet İşleri Başkanlığı, Din İşleri Yüksek Kurulu