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It would be "linguam latīnam" (accusative). as in "linguam latīnam discimus (we learn the latin language). But "studēre" doesn't really have the same meaning as "to study".
you should understand it as "we dedicate ourselves TO the latin language". with the dative (linguae latīnae) having the function of the word "to".
Thank you very much for this. I was stuck on a similar phrase in the "School" section and I couldn't understand why it wasn't in the accusative. "Discipuli in ludo linguae Latinae student". It's the dative case! I've had the idea to use the word studious to help me remember eg. we are studious to the Latin language. Ok it might not be great English but it puts the word 'to' in there so I'll remember it's the dative case!
For me, this evokes the English expression, "I am a student of X" where X could be any inanimate object, concept, or phenomenon, and it means essentially "I am dedicated to the study of X."
Of course, "of X" is the English equivalent of genitive case, so it's clearly not a perfect parallel...
"We study Latin at home" ought to be accepted. That is the most common way of saying it in English, although it is not wrong to say "We study the Latin language at home".
In Latin, however, you must say "Linguae Latinae domi studemus" because in Latin, that's not the name of the language, that's just an adjective that modifies the word "language", saying which language it is.