Foretaksregisteret og Merverdiavgiftsregisteret er offentlige.

Because -et is the usual definite ending for a singular neuter noun in Norwegian.

• et register = a register
• registeret = the register

Both Foretaksregisteret and Merverdiavgiftsregisteret are compound nouns whose last part is register, so the definite ending is added at the end of the whole compound.

In Norwegian, the definite article is usually attached to the noun as a suffix instead of being a separate word.

So instead of saying something like the register, Norwegian normally says:

• registeret = the register

That is why you do not see a separate word like English the in this sentence.

Each noun by itself is singular, but when you join two singular nouns with og (and), the whole subject becomes plural.

So:

• Foretaksregisteret = one register
• Merverdiavgiftsregisteret = one register
• Foretaksregisteret og Merverdiavgiftsregisteret = two registers together

Because the subject is plural, the adjective also appears in its plural form:

• singular: offentlig
• plural: offentlige

Compare:

• Foretaksregisteret er offentlig.
• Foretaksregisteret og Merverdiavgiftsregisteret er offentlige.

In Norwegian, verbs do not change for singular and plural the way they do in English.

So er means is or are, depending on the subject.

Examples:

• Registeret er offentlig. = The register is public.
• Registrene er offentlige. = The registers are public.

The verb stays er in both cases.

They are compound nouns, which are extremely common in Norwegian.

Foretaksregisteret

This can be broken down as:

• foretak = business / enterprise / undertaking
• register = register
• -et = the

So the whole word means something like the business register or, more specifically, the Register of Business Enterprises.

Merverdiavgiftsregisteret

This can be broken down as:

• merverdiavgift = value-added tax / VAT
• register = register
• -et = the

So this means the VAT register.

Norwegian often builds long precise nouns by combining smaller words into one word.

That -s- is a linking sound used in many Norwegian compounds. It is often called a linking s.

So:

• merverdiavgift
  • register
• becomes merverdiavgiftsregister

This -s- is not a possessive ending here. It just helps connect the parts of the compound in a natural way.

You will see this a lot in Norwegian compound words.

It can sometimes have different shades of meaning depending on context, but here it means public in the sense of open to the public or publicly accessible.

In other contexts, offentlig can also refer to things connected with the public sector or government, such as:

• offentlig sektor = the public sector
• offentlige tjenester = public services

So the exact nuance depends on context, but in this sentence public is the natural meaning.

They are being treated as the names of specific official registers, so capitalization is natural.

In Norwegian, ordinary common nouns are usually not capitalized, unlike in English titles. But names of specific institutions, registers, agencies, and similar entities are often capitalized.

So:

• et register = a register
• Foretaksregisteret = the specific register called Foretaksregisteret

No, not in standard Bokmål with this plural subject.

Because the subject contains two things joined by og, the adjective should be plural:

• correct: Foretaksregisteret og Merverdiavgiftsregisteret er offentlige.
• not standard here: ... er offentlig.

If there were only one register, then offentlig would be correct:

• Foretaksregisteret er offentlig.

The sentence follows the normal Norwegian main-clause order:

subject + verb + adjective

• Foretaksregisteret og Merverdiavgiftsregisteret = subject
• er = verb
• offentlige = adjective/predicative complement

This is very similar to English:

• The Register of Business Enterprises and the VAT Register are public.

So the structure is quite straightforward for an English speaker.