Ég gleymdi tannkreminu heima.

Because gleyma takes its object in the dative case.

The noun is tannkrem (toothpaste), a neuter noun.
With the definite article:

• tannkremið = the toothpaste in the nominative/accusative
• tannkreminu = the toothpaste in the dative

Since Icelandic says literally something like I forgot the toothpaste with toothpaste as a dative object after gleyma, you get:

• Ég gleymdi tannkreminu.

So this is one of those verbs where you have to learn which case the object takes.

Yes, when gleyma means to forget something, the thing forgotten is normally in the dative.

Examples:

• Ég gleymdi lyklunum. = I forgot the keys.
• Hún gleymdi símanum. = She forgot the phone.
• Við gleymdum nafninu. = We forgot the name.

This is very important for learners, because English does not show this kind of case change, but Icelandic does.

Gleymdi is the past tense of gleyma (to forget).

So:

• að gleyma = to forget
• ég gleymi = I forget / I am forgetting
• ég gleymdi = I forgot

In this sentence, gleymdi clearly tells you the action happened in the past.

Heima means at home.

It refers to being in a place, not moving toward it.

So:

• heima = at home
• heim = home, homewards / to home

Compare:

• Ég er heima. = I am at home.
• Ég fer heim. = I am going home.

So in Ég gleymdi tannkreminu heima, heima means the toothpaste was left/forgotten at home.

Because the sentence describes location, not movement.

You are not going home in this sentence. You are saying where the toothpaste was forgotten: at home.

That is why Icelandic uses:

• heima for being somewhere
• heim for going somewhere

So:

• Ég gleymdi tannkreminu heima. = I forgot the toothpaste at home.
• Ég fór heim. = I went home.

It is singular.

Tannkrem is usually treated like an uncountable substance noun in this kind of sentence, just like toothpaste in English.

So:

• tannkrem = toothpaste
• tannkremið = the toothpaste
• tannkreminu = the toothpaste (dative)

Even though English sometimes uses mass nouns without an article, Icelandic often uses the definite form when referring to a specific item you mean in context.

Yes, you could say Ég gleymdi tannkremi heima, but it would sound more indefinite: I forgot toothpaste at home or I forgot some toothpaste at home.

The sentence Ég gleymdi tannkreminu heima sounds more natural if you mean a specific toothpaste you had in mind, such as the toothpaste we were supposed to bring.

So the difference is roughly:

• tannkreminu = the toothpaste
• tannkremi = toothpaste / some toothpaste

Both are dative, but one is definite and one is indefinite.

It is an adverb.

It tells you where something happened:

• Ég gleymdi tannkreminu heima.

So grammatically, heima is not the object of the verb. The object is tannkreminu, and heima just adds location.

This is a very normal Icelandic word order:

• Ég = subject
• gleymdi = verb
• tannkreminu = object
• heima = adverb of place

So it follows a simple pattern like:

Subject + Verb + Object + Adverb

That makes it a very natural basic sentence for a learner.

You may also hear different word orders in Icelandic depending on emphasis, but this version is straightforward and neutral.

Yes. Icelandic allows some flexibility, especially with adverbs and emphasis.

For example, you might also hear:

• Ég gleymdi heima tannkreminu.

But this sounds less neutral and may feel more marked depending on context.

For learners, Ég gleymdi tannkreminu heima is the safest and most natural order to use.

Normally, yes, you should say it.

Icelandic usually does not drop the subject pronoun the way Spanish or Italian often can.

So:

• Ég gleymdi tannkreminu heima. = correct, normal
• Gleymdi tannkreminu heima. = only works in special contexts, not as the normal full sentence

For a standard sentence, keep Ég.

You usually learn noun gender together with the noun.

Tannkrem is a neuter noun, and you can see that in its definite forms:

• tannkremið = the toothpaste
• tannkreminu = the toothpaste (dative)

The endings -ið and -inu are typical neuter definite endings in these forms.

In Icelandic, gender matters because it affects:

• articles
• adjective endings
• pronouns
• case forms

So it is best to memorize nouns with gender from the start.

Usually, yes.

Heima is the normal everyday word for at home, and that is exactly what you want in most situations.

It often overlaps with what English speakers mean by at my house or back at home, depending on context.

If you want to be more explicit, you can say something like:

• Ég gleymdi tannkreminu heima hjá mér. = I forgot the toothpaste at my place / at home

But in ordinary speech, heima is very common and natural.