I would like to know if it is necessary for a crucifix to be present at the altar in order for the Mass to be valid.
Masses are not valid or invalid, they are licit (in conformity with the law) or illicit (not in conformity with the law). It is the consecration of the Eucharist within the Mass that can be valid or invalid.
Whether a crucifix is present at the altar has no bearing on whether the Eucharist is valid or invalid (the absence of a crucifix will not cause the Eucharist to be invalid), but it does have a bearing on whether the Mass is licit or illicit.
The law requires that “There is to be a cross, clearly visible to the congregation, either on the altar or near it” (General Instruction of the Roman Missal [GIRM] 270). The revised General Instruction—which has not yet gone into effect—clarifies that the cross in question should have a corpus (representation of Christ’s body), meaning that it should be a crucifix rather than a bare cross.
If there were no cross by on or near the altar (or, once the new GIRM goes into effect, no crucifix) then the Mass would be illicit, or not celebrated in accord with the requirements of the law.
However, a Mass celebrated in this manner would still have a valid consecration of the Eucharist. Furthermore, it would still fulfill one’s Sunday obligation.