Mater filiae aegrotae adest.

Adest is from adsum, a compound of sum (to be) with ad (to, near, at).

So est just means is, but adest has a stronger sense like:

• is present
• is here
• is at hand
• sometimes is there for someone

In this sentence, adest is important because it often goes with a dative noun, showing the person someone is present to or for.

The subject is mater.

You can tell because:

• mater is in the nominative singular
• adest is third person singular
• the verb must agree with its subject, so a singular subject fits

So the basic structure is:

mater ... adest = the mother ... is present/is here

Mater is the nominative singular form of mater, matris (mother).

It belongs to the third declension.
That can surprise English speakers, because they may expect a feminine noun to look like a first-declension noun such as filia. But mater is one of several common family words in the third declension, like:

• mater, matris = mother
• pater, patris = father
• frater, fratris = brother

So mater is nominative here because it is the subject.

Here it is best understood as dative singular:

• filiae = to/for the daughter
• aegrotae = sick, agreeing with filiae

This fits the verb adest, which commonly takes a dative: someone is present to/for someone.

So the phrase means to/for the sick daughter.

Because aegrotae is an adjective modifying filiae, and Latin adjectives must agree with the nouns they describe in:

• gender
• number
• case

Here both words are:

• feminine
• singular
• dative (in this interpretation)

So they both take the same ending:

• filiae
• aegrotae

This is normal in Latin. The matching endings show that the adjective belongs with that noun.

Because aegrotae matches filiae, not mater.

If aegrota described mater, we would expect mater aegrota:

• mater = nominative singular
• aegrota = nominative singular feminine

But the sentence has aegrotae, which matches filiae in case, number, and gender.

So aegrotae goes with filiae, not with mater.

Yes, it can be.

The form filiae aegrotae could grammatically be:

• dative singular = to/for the sick daughter
• genitive singular = of the sick daughter
• in isolation, some -ae forms can also be plural, though that does not fit this sentence well

So strictly speaking, the phrase by itself is ambiguous.

However, in this sentence the verb adest strongly favors the dative interpretation, because adsum often takes the dative of the person someone is present to or for.

So although there is form ambiguity, the syntax helps resolve it.

Classical Latin has no definite article like English the and no indefinite article like a/an.

So mater can mean:

• mother
• the mother
• sometimes even a mother

The exact sense depends on context.

English must choose an article, but Latin usually leaves that unstated.

Latin word order is much more flexible than English word order, because the endings show the grammatical roles.

So all of these can express the same basic idea:

• Mater filiae aegrotae adest
• Mater adest filiae aegrotae
• Filiae aegrotae mater adest

The chosen order often affects emphasis, not the core grammar.

In your sentence, placing mater first gives it prominence, and ending with adest gives a neat verbal finish.

Because the subject is singular: mater.

Even though there are three words before the verb, only mater is the subject.
Filiae aegrotae is a dependent phrase, not another subject.

So the sentence uses:

• adest = he/she/it is present

not a plural form such as adsunt.

They are:

• mater, matris = mother
• filia, filiae = daughter
• aegrotus, aegrota, aegrotum = sick, ill
• adsum, adesse, adfui = be present, be here, be near

A learner often needs the dictionary form because the form in the sentence may not look exactly like the headword. For example:

• adest comes from adsum
• aegrotae comes from aegrota / aegrotus
• mater is already the dictionary nominative singular form, but its genitive matris shows its declension pattern