Við höfum unnið að rannsókninni saman.

Breakdown of Við höfum unnið að rannsókninni saman.

við
we
saman
together
hafa
to have
vinna að
to work on
rannsóknin
the research
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Questions & Answers about Við höfum unnið að rannsókninni saman.

What tense is höfum unnið, and how do you form it?
That’s the present perfect (núliðin tíð). It combines the present tense of hafa (“to have”) – here höfum (“we have”) – with the past participle unnið of að vinna (“to work”).
Why do we use vinna að instead of just vinna here, and what role does the preposition play?
In Icelandic “to work on something” is vinna að einhverju. The preposition (“on/to”) governs its object in the dative case. So you need vinna að rannsókninni to say “work on the research.”
Why is the noun rannsókninni in that specific form instead of rannsókn?
Icelandic attaches the definite article as a suffix to the noun. Rannsókn is a feminine noun. The dative singular definite ending is -inni, so rannsókninni means “to/for the research” (definite, dative).
Which grammatical case is rannsókninni, and why do we use that case here?
It’s the dative singular definite. The preposition used with vinna requires its object to be in the dative, hence rannsókninni.
What does saman mean, and why is it placed at the end of the sentence?
Saman means “together.” In Icelandic, adverbs and adverbial modifiers often come after the main verb or verb phrase, especially in a simple statement like this. That’s why saman ends up at the end.
Can you place saman somewhere else, like between höfum and unnið, or does that sound odd?
The most natural placement is after the verb phrase: Við höfum unnið saman að rannsókninni or at the very end as in the example. Putting saman between höfum and unnið (Við höfum saman unnið…) splits the auxiliary from its participle and sounds awkward.
Is there a distinction between the simple past við unnum and the perfect tense við höfum unnið?
Yes, though subtler than in English. Við unnum (preterite) simply states a past action. Við höfum unnið (perfect) often implies relevance to the present or a continuing state. Here, við höfum unnið að rannsókninni suggests the work has been ongoing up to now.
How would you express “we are working on the research together” in Icelandic (the continuous aspect)?

Use the vera að + infinitive construction for the progressive:
Við erum að vinna að rannsókninni saman.
Here erum is the present of vera (“to be”), að vinna the infinitive, plus saman as before.