So I have the following sentence which I have to translate into Latin:
The farmer gives his daughter water.
The parts which I found easy: Agricola ... aquam dat.
I don't know how to express "his" because I never really learned it at this point.
My book gives: Agricola filiae aquam dat.
Can somebody explain me what is happening? Wouldn't this translate to: The farmer of the daughter gives water.
The word filiae can be either genitive or dative; the two cases look alike in the singular of the first declension. If you read it as a genitive, then filiae is indeed "of the daughter" and this translation is correct:
Agricola filiae aquam dat.
The farmer of the daughter gives water.
If you read it as a dative, then filiae means "to the girl", leading to a different translation:
Agricola filiae aquam dat.
The farmer gives water to the daughter.
They are both equally correct readings of the Latin sentence. Only context determines which is correct. The second one makes more sense here; "farmer of the daughter" sounds unusual.
In your book the intended translation is the second one I give. But it does not specify the relation of the farmer and the daughter at all. You can add a "his", but Latin behaves very differently to English here. The word suus is an adjective modifying the daughter, not a genitive of the farmer. It has to have the same number, gender and case than the daughter. This earlier question and its answers might be a useful read.
Thus the most natural translation would be:
The farmer gives his daughter water.
Agricola filiae suae aquam dat.
The general word for "his" (or "hers", or "its", or "theirs") in Latin is eius. This is the genitive singular of is/ea/id, "he/she/it". Those are three separate words, but conveniently they all share a genitive singular form.
However, I wouldn't use eius in this case, when the farmer has already been mentioned in the sentence. Because Latin uses reflexive pronouns slightly differently than English does.
In English, a reflexive pronoun (like "himself", "myself", "themselves") can occur for any person, but only in the accusative. In Latin, reflexives appear only in the third person (there's no equivalent to "myself": just use "me"). But they can appear in any case besides the nominative, and there are even possessive reflexives, sort of like English "his own" or "belonging to himself".
This is one of the instances where a possessive reflexive is exactly what you want, since the farmer's also the subject of the verb. The word for this is suus, sua, suum.
You decline it pretty much like a normal adjective. But there's yet another catch: in English, the gender of a possessive is the gender of the possessor (he gave it to his daughter). But in Latin, the gender of a possessive is the gender of the thing being possessed.
So you might see suus fīlius "his son", or suus fīlius "her son", but NOT *sua fīlius. The son is masculine, so suus must be masculine, no matter whose son it is.
In this case, you would want suæ fīliæ, "his own daughter".
This does look very much like a genitive singular ("...of his own daughter"). But for first-declension nouns, the genitive singular and dative singular are both marked by -æ, and you need context to know which. Since in this case you have the verb dāre, which requires a dative somewhere, it's reasonably certain that fīliæ is dative rather than genitive. The suæ also makes this clear, since "the farmer of his own daughter" sounds very wrong.
And now that we've told you all of this...you don't always need it. Latin is much more forgiving of leaving off pronouns than English is. Just as fīliæ on its own could be translated as "a daughter" or "the daughter", it could also be translated as "his daughter", with a suæ just implied. That's what the book's answer does: it's most likely that the farmer is giving water to his own daughter, rather than someone else's, so there's no need to say it explicitly.
