Invidia vulgi, quod tribus militibus fortuna publica commissa fuerit, vanum ingenium dictatoris corrupit.
What kind of subjunctive is fuerit and why. What tense is corrupit — perfect with or without have? Why?
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Fuerit is the perfect subjunctive of esse, you can look up such forms in a conjugation table like this. Part of learning Latin is guessing what form of what verb you might be looking at, and then confirming your guess in the dictionary if you do not know the verb by heart. This is a nice thing about printed dictionaries: Even if you do not find corrupi…, you scan over the page and quickly notice corrumpo.
With that said, if you need help identifying verb forms, I recommend Wiktionary as a resource. Whatever else their qualities, they allow you to look up a form like corrupit and the page will tell you exactly what it is.
But the source of your confusion is probably something else: What in the world is the perfect of esse doing there as an auxiliary verb? There are quite nice Latin grammars that never mention a single word about this possibility. But that is where this website comes to your help with this question, which should clear up any confusion. But to quote the most pertinent part of cnread's answer (quoting several reputable grammars):
The fui, etc., forms are rarely found in CICERO, never in CAESAR, but are characteristic of LIVY and SALLUST. […] Similarly in the Perfect and Pluperfect Subjunctive fuerim, fueris, etc., are sometimes used instead of sim, sis, etc., and fuissem instead of essem.
In case you wonder why there is a subjunctive at all: This indicates that the reason given for the mob's envy is subjective, that is, the author is not endorsing this point of view, but it was the reason the people were envious, or which they may have cited as the reason for their envy. If the author had instead written:
Invidia vulgi, quod fortuna publica commissa fuit …
then he would have presented the reason as an objective fact – the author agrees that the public fate had been entrusted, and believes that this is what made the mob envious.
This distinction, which exists not only with quod but also with quia and other causal conjunctions, can easily be lost in the English translation if you choose a plain "because" sentence, like so: “The envy of the masses, because the public fate was entrusted to three military men, …” As a translator, you may make the judgement call that this is fine, as rendering the subjective character in English can be awkward. But not rendering it can also be awkward in some cases! Take this example:
Socrates capitis damnatus est, quia iuvenes corrumperet.
How would you translate this? Surely not like this: “Socrates was sentenced to death because he was corrupting the youth.” That would be all right if the sentence had read: quia iuvenes corrumpebat. But it was corrumperet. The Latin original does not present the charge against Socrates as an objective fact, and the translation should preserve this, but that requires you as a translator to choose a freer translation, such as: “Socrates was sentenced to death based on the charge that he was corrupting the youth.”