Eftir rigninguna sáum við fallegan regnboga yfir höfninni.

Eftir means after here, in a time sense: after the rain.

The noun rigninguna is the accusative singular definite form of rigning (rain). That happens because eftir takes the accusative when it means after in expressions of time.

So:

• rigning = rain
• rigninguna = the rain, in accusative

A learner-friendly way to read it is:

• Eftir rigninguna = After the rain

This is because Icelandic normally follows the verb-second rule in main clauses.

If the sentence starts with something other than the subject, the finite verb usually still comes in the second position.

Here the sentence begins with the time phrase:

• Eftir rigninguna = after the rain

So the verb comes next:

• sáum = saw

And then the subject follows:

• við = we

That gives:

• Eftir rigninguna sáum við ...

If you started with the subject instead, you could say:

• Við sáum fallegan regnboga eftir rigninguna.

Both are possible, but the original sentence puts emphasis first on after the rain.

Sáum is the past tense, 1st person plural form of sjá (to see).

So:

• sjá = to see
• við sáum = we saw

A few useful past-tense forms are:

• ég sá = I saw
• þú sást = you saw
• hann/hún/það sá = he/she/it saw
• við sáum = we saw

So in this sentence, sáum við simply means we saw.

Because regnboga is the direct object of the verb sáum (we saw), and sjá takes an object in the accusative.

The dictionary form is:

• regnbogi = rainbow

But here it changes to the accusative singular:

• regnboga

The adjective has to agree with the noun in gender, number, and case:

• fallegur = beautiful
• fallegan = beautiful, masculine singular accusative

So:

• fallegan regnboga = a beautiful rainbow

Because yfir can take different cases depending on the meaning.

A very common rule is:

• accusative = movement toward/across something
• dative = location above/over something

In this sentence, the rainbow is simply located over the harbor, not moving across it. So Icelandic uses the dative:

• yfir höfninni = over the harbor

Compare:

• yfir höfninni = over the harbor, in a position above it
• yfir höfnina = over/across the harbor, with movement

So the sentence uses höfninni because it describes where the rainbow was.

In Icelandic, the definite article is usually attached to the end of the noun instead of being a separate word.

So:

• höfn = harbor
• höfnin = the harbor

And when the noun changes case, the definite form changes too:

• höfninni = the harbor in dative singular

Likewise:

• rigning = rain
• rigningin = the rain
• rigninguna = the rain in accusative singular

So Icelandic often expresses the as a suffix, not as a separate word like English does.

Because the sentence is introducing a beautiful rainbow, not the beautiful rainbow.

In Icelandic, just like in English, you can talk about something as:

• indefinite = a rainbow
• definite = the rainbow

Here it is indefinite:

• fallegan regnboga = a beautiful rainbow

If you wanted the beautiful rainbow, you would use a definite noun form, and the adjective form would also change:

• fallega regnbogann = the beautiful rainbow

So the sentence uses the indefinite form because the rainbow is being mentioned as something newly seen, not as a specific already-known one.

That ending shows agreement with the noun.

Here, regnbogi is:

• masculine
• singular
• accusative
• indefinite

So the adjective takes the matching strong ending:

• fallegan

This is part of a larger Icelandic pattern:

• adjective + noun must match in gender
• adjective + noun must match in number
• adjective + noun must match in case

That is why Icelandic adjectives change much more than English adjectives do.

You would make the noun definite, and the adjective would change to the weak form.

So:

• fallegan regnboga = a beautiful rainbow
• fallega regnbogann = the beautiful rainbow

If we replace that part in the sentence, we get:

• Eftir rigninguna sáum við fallega regnbogann yfir höfninni.

This is a very important Icelandic pattern:

• indefinite noun → usually strong adjective
• definite noun → usually weak adjective

So this sentence is a good example of the indefinite pattern.

A useful breakdown is:

• Eftir rigninguna = time expression
• sáum = verb
• við = subject
• fallegan regnboga = direct object
• yfir höfninni = place expression

So the sentence is built like this:

• After the rain
  • saw
    • we
      • a beautiful rainbow
        • over the harbor

Even though that looks unusual from an English point of view, it is very natural in Icelandic because of the verb-second rule and the use of case endings to show each word’s role.