When Latin nouns are listed for memorisation they are listed with the nom. sg., the gen. sg. and their gender. E.g. agricola, agricolae, masculine. Why are each of these forms necessary for memorisation? Can they not be worked out from each other?
Can they not be worked out from each other?
No! That's the whole point for learning all of them.
Why are each of these forms necessary for memorisation?
That is the minimal amount of information to deduce how to use the noun.
To be able to use a noun, you need to know all its forms and its gender. You cannot reliably deduce the gender from the forms, although one can often guess well and there are some clear cases. In fact, the singular nominative is the least necessary piece of information out of the three, but it is sometimes necessary and in my opinion always useful.
Here are some examples:
The words agricola and traha are declined exactly the same way: agricolam/traham, agricolae/trahae and so on. But they are different gender, and there is no way you can tell that by looking at the forms.
There is a way to guess these right: A sledge (traha) does not have a natural gender, so it is feminine like most words of the first declension. A farmer is primarily male, so agricola is masculine instead. But sometimes you would guess wrong, and you should not rely on guesswork. That's why you should learn the gender and why a dictionary will give it to you.
The words agricola, problema, and castra all seem to end in -a. But based on that alone you can't figure out the other forms. With the genitives — agricolae, problematis, castrorum — you know much more. The declensions are first, third, and second, and castra is plural. (The singular castrum exists also but is rarer.)
If you have learned the case endings well, you will recognize that castra has to be neuter, and it is indeed. As discussed earlier, agricola could be feminine or masculine based on form. But with a third declension word like problema you cannot tell the gender by the nominative and the genitive. And you cannot tell the singular accusative and plural nominative and accusative without knowing the gender. Therefore a dictionary will tell you that it's neuter.
It helps if you remember that all words in -ma, gen. -matis, are neuter (and Greek loans). But you don't have to know such things. Just memorize the gender with this word or look it up in a dictionary.
To give a single fuller example, you need to know both the genitive and the gender to figure out the singular accusative of problema. It is not problemam or problematem but problema.
Joonas's answer is spot-on, but to give some more illustrations:
Quite a lot of Latin nouns end in -us; it's one of the best-known features of the language. But they don't all decline the same!
- Servus "slave", genitive servī
- Tempus "time", genitive temporis
- Scelus "crime", genitive sceleris
- Manus "hand", genitive manūs
You need the genitive to know which of these patterns it follows.
You also need the gender to deal with adjectives, because (most) adjectives reflect the gender of the noun they're modifying, not the declension pattern. These three nouns all follow the -us, -ī pattern, but have different genders:
- Servus bonus "a good slave", masculine
- Quercus bona "a good oak", feminine
- Vulgus bonum "a good crowd", neuter
All three of these decline like servus (with a genitive in -ī), but their genders are different.
And unfortunately you can't always derive the nominative from the genitive, either:
- Hominis "of the person", nominative homō
- Nōminis "of the name", nominative nōmen
- Turris "of the tower", nominative turris
- Tūris "of the incense", nominative tūs
- Honōris "of honor", nominative honor
- Floris "of the flower", nominative flos
- Corporis "of the body", nominative corpus
So in the end, you need all three of the nominative, genitive, and gender; no one or two of those will suffice.