Við sáum radísum í ræktunarreitinn.

Not quite. A native-English-speaking learner would be right to pause here, because two forms look unusual:

• radísum is dative plural
• ræktunarreitinn is accusative singular with the definite article

With the verb sjá (to see), the thing seen is normally in the accusative, so you would usually expect radísur, not radísum.

Also, after í, Icelandic uses:

• dative for location: in
• accusative for motion into something: into

Since seeing does not itself express movement, if the meaning is We saw radishes in the garden bed / cultivation plot, you would normally expect:

• Við sáum radísur í ræktunarreitnum.

So this sentence is useful for learning, but as written it sounds off unless there is some very unusual context.

Við means we.

Sáum is the past tense of sjá (to see) for we. So:

• sjá = to see
• við sáum = we saw

The ending -um is part of the past-tense conjugation for we in this verb. Icelandic verbs change form depending on person and number, much more than English verbs do.

For example:

• ég sá = I saw
• þú sást = you saw
• við sáum = we saw

So the verb ending tells you who performed the action.

Because radísum is the dative plural form of radísa (radish), but the verb sjá normally takes a direct object in the accusative.

A learner would usually expect:

• radísur = radishes (plural, accusative/nominative form) not
• radísum = to/for/involving radishes (dative plural)

So if the intended meaning is simply We saw radishes..., the expected form is:

• Við sáum radísur ...

This is a very common issue for English speakers, because English nouns usually do not change form for case, while Icelandic nouns do.

Normally, sjá takes the accusative.

So:

• að sjá radísu = to see a radish
• að sjá radísur = to see radishes

That means:

• singular accusative: radísu
• plural accusative: radísur

If the sentence means We saw radishes, then radísur is the expected object form.

This is one of the most important preposition patterns in Icelandic.

With í:

• dative = location, in
• accusative = motion/direction, into

Examples:

• í húsinu = in the house
• í húsið = into the house

So if you are talking about where something already is, use dative. If you are talking about movement into that place, use accusative.

In your sentence, í ræktunarreitinn uses the accusative, which suggests into the cultivation plot / bed. But with We saw..., that does not fit very naturally unless some movement idea is intended separately.

You would expect the dative after í, because this is location rather than motion.

So:

• í ræktunarreitnum = in the cultivation plot / bed

That gives the more natural sentence:

• Við sáum radísur í ræktunarreitnum.

This means We saw radishes in the cultivation plot/bed.

Ræktunarreitinn is a compound noun with the definite article attached.

It can be broken down like this:

• ræktun = cultivation, growing
• reitur = plot, patch, bed, square of land
• ræktunarreitur = cultivation plot / growing bed / garden bed
• ræktunarreitinn = the cultivation plot / the garden bed (accusative singular)

The -inn at the end is the attached definite article, meaning the.

This is very common in Icelandic: instead of a separate word like English the, Icelandic often adds the article to the end of the noun.

Because Icelandic usually expresses the as a suffix attached to the noun.

Compare:

• ræktunarreitur = a cultivation plot
• ræktunarreiturinn = the cultivation plot

In this sentence, because the noun is in the accusative singular, the form is:

• ræktunarreitinn

English speakers often find this unfamiliar at first, but it is a basic feature of Icelandic grammar.

Yes, Icelandic word order is somewhat more flexible than English, but not completely free.

The basic order here is:

• Við = subject
• sáum = verb
• radísur = object
• í ræktunarreitnum = prepositional phrase

A natural version would be:

• Við sáum radísur í ræktunarreitnum.

You could also move parts around for emphasis in some contexts, but for a learner, the basic order above is the safest and most natural.

If the intended meaning is We saw radishes in the cultivation plot / garden bed, the most likely corrected version is:

• Við sáum radísur í ræktunarreitnum.

Why?

• radísur = accusative plural object after sáum
• í ræktunarreitnum = dative after í for location

So the original sentence is a good opportunity to notice two very important Icelandic patterns:

1. verbs can require a specific case on their object
2. í changes case depending on location vs motion