"They do the course about French culture."
Translation:Déanann siad an cúrsa faoi chultúr na Fraince.
Could you not say, "faoi chultúr fhrancach"? I tried it, and it wasn't accepted. Only "faoi chultúr na Fraince" seems to work, but why? Do we take this to mean "the culture of France?
the French king: rí na Fraince (the King of France - makes sense to me)
French lavender: labhandar francach French window: fuinneog fhrancach
And these use "Fraincise" because it relates to the language, French (Fraincis): French teacher: múinteoir Fraincise French lesson: ceacht Fraincise French book: leabhar Fraincise
One of the uses of the tuiseal ginideach is to turn a noun into an adjective, so cultúr na Fraince can be read as "the culture of France* or "French culture".
As far as I know, the TG is preferred if you can reasonably say "x of y". "king of France" and "culture of France" use the na Fraince constructin, but "lavendar of France" and "window of France" don't make sense, so they use the adjectival form francach, and a "teacher of French" is múinteoir Fraincise, but a teacher who works in France is a múinteoir Francach, even though they are both a "French teacher" in English.
But don't forget that in the Irish language (with a few exceptions) the name of a country always has the definite article with it. For example, "I like France" would be "Is maith liom an Fhráinc" even though the English sentence has no definite article. So for this exercise, a definite article shows up not because of culture, but because of France.