Learn Hardcore Hungarian: A diák néha otthon felejti a könyvet. - Sometimes the student forgets the book at home.

Hungarian usually repeats the article in front of each definite noun phrase, so you say A diák nézi a könyvet (The student is looking at the book), not A diák nézi könyvet.

You only leave it out if you want to say something more general, like diák = a student / students (in general), or könyvet (without article) = a book / books (in general).

Why is it felejti and not felejt?

Hungarian verbs have two present-tense conjugations: indefinite and definite.

felejt = 3rd person singular, indefinite (no specific, clearly defined direct object)
felejti = 3rd person singular, definite (there is a specific direct object)

In the sentence, a könyvet (the book) is a definite object, so you must use the definite form of the verb:

A diák néha otthon felejti a könyvet.
→ The student sometimes forgets the book at home.

If the object were indefinite, e.g. könyvet without a, you would normally use the indefinite form:

A diák néha otthon felejt könyvet.
→ The student sometimes forgets (some) book(s) at home. (less specific)

What is the infinitive of felejti, and how is it formed?

The infinitive is felejteni = to forget.

From felejteni, the stem is felejt-. In the present tense:

(ő) felejt = he/she forgets (indefinite)
(ő) felejti = he/she forgets it / the X (definite)

So felejti is built from the same stem felejt- plus the definite ending -i for 3rd person singular.

Why is it könyvet and not just könyv?

Könyv means book (dictionary form).

When könyv is a direct object, Hungarian usually marks it with the accusative -t:

könyv → könyvet (book → book as object)

The extra -e- appears for phonetic reasons: könyvt would be hard to pronounce, so Hungarian inserts e to break it up: könyv + t → könyvet.

So:

a könyv = the book (subject or standalone noun)
a könyvet = the book (as the object of a verb)

Why is otthon used without any ending, when English says at home?

Otthon is a special adverb meaning at home.

You don’t say at separately in Hungarian; otthon already includes the idea of at. So:

otthon = at home
Otthon vagyok. = I am at home.
Otthon felejti a könyvet. = He/She forgets the book at home.

If you add a case ending, you change the meaning slightly:

otthonról = from home
otthonra = to home (towards home)

But for at home, you just use bare otthon.

Where can néha go in the sentence, and does its position change the meaning?

Néha means sometimes and is quite flexible:

A diák néha otthon felejti a könyvet.
A diák otthon néha felejti a könyvet.
Néha a diák otthon felejti a könyvet.

All are grammatical. The basic meaning (The student sometimes forgets the book at home) stays, but word order in Hungarian also reflects focus and emphasis.

The position right before the verb is the strongest focus position.
If you put néha earlier (e.g. at the very beginning: Néha a diák…), you’re putting more emphasis on sometimes = It is sometimes that…
If otthon is directly before the verb (A diák néha otthon felejti…), then otthon is more strongly emphasized (forgetting at home rather than elsewhere).

For everyday use, A diák néha otthon felejti a könyvet is neutral and natural.

Could we say A diák néha a könyvet otthon felejti? Is that correct?

Yes, it is correct:

A diák néha a könyvet otthon felejti.

This word order gives a bit more emphasis to a könyvet by placing it closer to the verb. It can sound like you’re contrasting this object with something else:

It’s the book that the student sometimes forgets at home (not something else).

Neutral, default word order is usually:

A diák néha otthon felejti a könyvet.

But the version with a könyvet closer to the verb is also perfectly good Hungarian, just slightly more focused.

Is diák singular or plural here, and how would the sentence look in plural?

In this sentence, diák is singular: the student.

For plural:

A diákok néha otthon felejtik a könyvet.
→ The students sometimes forget the book at home. (one specific book, shared context)

If you mean each of them forgetting their own book, you can also pluralize the object:

A diákok néha otthon felejtik a könyveket.
→ The students sometimes forget the books at home.

Note that the verb also becomes plural:

felejti (he/she forgets it) → felejtik (they forget it/them)

Why does Hungarian use the present tense here, when English might say sometimes forget or sometimes leaves?

Hungarian present tense covers both English simple present and present continuous, depending on context.

A diák néha otthon felejti a könyvet.
Literally: The student sometimes forgets the book at home.
This matches the English simple present used for habits.

Hungarian does not normally distinguish is forgetting vs forgets in form; context and adverbs like néha (sometimes), mindig (always), etc., show that it’s a habitual action.

Could we use hagyni instead of felejteni here? What’s the difference?

felejteni = to forget (you unintentionally don’t take something with you, or it slips your mind)
hagyni = to leave (something somewhere), usually more neutral about intention

So:

A diák néha otthon felejti a könyvet.
→ The student sometimes forgets the book at home. (carelessness, unintentional)
A diák néha otthon hagyja a könyvet.
→ The student sometimes leaves the book at home. (could be on purpose or neutral)

Both are correct, but felejti highlights the idea of forgetting.

Why is it felejti a könyvet and not felejti a könyv?

Because a könyvet is the object of felejti, it must be in the accusative case, marked by -t:

a könyv = the book (subject / base form)
a könyvet = the book (object)

Hungarian usually requires the -t ending when a noun is the direct object of a verb. So:

Látom a könyvet. = I see the book.
Olvasom a könyvet. = I read the book.
Felejtem a könyvet. / Felejti a könyvet. = I forget the book. / He/She forgets the book.

Is néha the only way to say sometimes here? Are there alternatives?

Néha is the most common neutral word for sometimes.

You could also use:

időnként = from time to time
- A diák időnként otthon felejti a könyvet.
alkalmanként = occasionally (more formal)
- A diák alkalmanként otthon felejti a könyvet.

All three keep the same grammar and overall structure; only the style and nuance change slightly.