Is it permissible to recite the Tashahhud without sending blessings upon the Prophet (peace be upon him) out of ignorance, and must the prayer be repeated?

The Tashahhud is not valid without saluting the Prophet (peace be upon him). The minimum sufficient wording is: "All greetings and acts of worship are for Allah. Peace be upon you, O Prophet, and Allah's mercy. Peace be upon us and upon the righteous servants of Allah. I bear witness that there is no god but Allah, and I bear witness that Muhammad is His Messenger." The preponderant opinion is that the Tashahhud is a pillar (rukn), so whoever omits it or performs it incompletely, their prayer is invalid. Some scholars have permitted not repeating the prayer for someone who omitted a pillar out of ignorance, and it is permissible to adopt a less preponderant opinion after the action has occurred. However, if you repeat the prayers in which you erred in the Tashahhud, that would be better.