Skolen begynder igen den 15. august, efter at ferien er slut.

Because Danish usually adds the definite article to the end of the noun.

• skole = school
• skolen = the school
• ferie = vacation / holiday
• ferien = the vacation / the holiday

So -en here is the equivalent of English the for common-gender nouns.

In this sentence, Danish uses the definite form skolen, literally the school.

That often happens when talking about a specific school context already understood by the speaker and listener, such as school starting again after the holiday. English often uses just school, but Danish more naturally uses skolen in sentences like this.

Danish often uses the present tense for future events when the time is clear from context.

Here, the date den 15. august makes it obvious that this is about the future, so begynder is completely natural.

This works much like English in sentences such as:

• School starts again on August 15
• My train leaves tomorrow

So begynder does not have to mean only present-time action.

Igen means again.

It shows that school is resuming after a break. In this sentence, it comes after the verb:

• Skolen begynder igen = School begins again

That position is very normal in Danish.

That is the normal Danish way to give a date in running text.

• den = the date marker
• 15. = fifteenth
• august = August

So den 15. august corresponds to English on the 15th of August or August 15.

In Danish, the number is written with a full stop in dates: 15.

Very often, yes, when giving a full date like this in a sentence.

• den 15. august is the standard form

Unlike English, Danish does not need a separate preposition like on here. The date expression itself works fine as written.

In Danish, names of months are normally written with a lowercase letter.

So:

• august
• september
• januar

This is different from English, where month names are capitalized.

Efter at means after when it introduces a whole clause.

Here it introduces:

• ferien er slut = the holiday is over

So:

• efter at ferien er slut = after the holiday is over

The at is part of the conjunction here and is very common in this structure.

Because this is a subordinate clause, and Danish word order changes in subordinate clauses.

In a main clause, Danish typically has the finite verb in second position:

• Ferien er slut

But after a conjunction like efter at, the clause becomes subordinate, and the usual order is:

• subject + verb
• ferien er slut

So ferien er slut is the normal subordinate-clause word order.

Here slut is being used as an adjective, meaning over, finished, or at an end.

So:

• ferien er slut = the holiday is over

By contrast, sluttet is the past participle of slutte and is used in other structures, such as passive or perfect-like constructions.

In this sentence, slut is the natural choice.

Because efter at ferien er slut is a subordinate clause.

In Danish, many writers place a comma before a subordinate clause like this. That is why you see:

• Skolen begynder igen den 15. august, efter at ferien er slut.

You may also see Danish written without this comma, depending on the comma system being used. So the version without the comma can also occur.

Yes. You could say:

• Efter at ferien er slut, begynder skolen igen den 15. august.

That is also correct.

Notice the word order in the main clause after the fronted subordinate clause:

• begynder skolen
• not skolen begynder

That happens because Danish main clauses follow the verb-second pattern: when something else comes first, the verb comes before the subject.

In dates, 15. is normally read as an ordinal:

• den femtende august

So although it is written 15., it is understood as the fifteenth.