Learn Hardcore Icelandic: Við viljum leysa þetta saman.

viljum = “we want.” It’s 1st person plural present of að vilja (to want).
Present tense of vilja:
- ég vil
- þú vilt
- hann/hún/það vill
- við viljum
- þið viljið
- þeir/þær/þau vilja
It behaves like a modal verb (“want to”) and is followed by an infinitive.

Why is it leysa and not a conjugated form like leysum?

After modal verbs like vilja, the next verb appears in the bare infinitive: viljum leysa.
Compare: Ég vil fara (I want to go), Þau vilja borða (They want to eat).

Where does negation go? How do I say “We don’t want to solve this together”?

What case is þetta, and why that form?

þetta is the neuter singular demonstrative “this/that.” Neuter nominative and accusative are both þetta.
Here it’s the direct object, so accusative, but neuter accusative = þetta anyway.
Other forms you’ll see: dative þessu, genitive þessa; plural nom/acc þessi.

Why Við and not Okkur?

við is nominative “we” (subject form). okkur is accusative/dative “us.”
Subject vs. object:
- Við leysum þetta. (We solve this.)
- Þetta hjálpar okkur. (This helps us.)
Genitive “our”: okkar.

Can I move saman to another position? What’s the usual word order?

Default and most natural: Við viljum leysa þetta saman.
You can front for emphasis (keeping verb-second): Saman viljum við leysa þetta. (Stylistic/emphatic)
Placing saman before the object is possible but marked: Við viljum saman leysa þetta (more formal/literary).
As an adverb of manner, saman often comes near the end of the clause.

How do I pronounce the sentence?

Is saman a preposition like “with”? When do I use með instead?

saman is an adverb meaning “together.” It does not take a case.
með is a preposition (“with”) and takes dative: með mér/þér/honum…
Together:
- We solve this together: Við leysum þetta saman.
- I solve this with you: Ég leysi þetta með þér.

Does leysa require any particular case for its object?

leysa is transitive and takes the accusative: leysa [acc].
- leysa þetta, leysa vandamál, leysa dæmið.
There are idiomatic expressions like leysa úr læðingi, but the basic verb takes a direct accusative object.

Should I say að before the infinitive? Is Við viljum að leysa… correct?

No að after modal vilja: use the bare infinitive: Við viljum leysa…
Use að to introduce a finite clause: Ég vil að þú leysir þetta. (I want you to solve this.)

How do I talk about past or future with this structure?

Icelandic often uses the present for future meaning if context makes it clear.
- “We want to solve this (now/soon).” Við viljum leysa þetta…
Past “wanted”: Við vildum leysa þetta saman.
- Preterite of vilja: ég vildi, þú vildir, hann vildi, við vildum, þið vilduð, þau vildu.

What’s the difference between viljum, ætlum, and skulum?

viljum = we want (desire).
ætlum = we intend/plan, often like “we’re going to”: Við ætlum að leysa þetta.
skulum = suggestion/let’s: Við skulum leysa þetta ≈ “Let’s solve this.”

Can I drop the subject pronoun like in Spanish? Could I say just “Viljum leysa þetta saman”?

No. Icelandic is not a pro-drop language. You must include the subject pronoun: Við viljum…
The exception is imperatives: Leysum þetta saman! (Let’s solve this together!)

How do I choose between þetta, það, and þessi?

þetta = “this/that” as a stand-alone neuter pronoun for an idea/thing just mentioned: leysa þetta.
það = “it/that,” also used as a dummy subject: Það er kalt.
þessi/þetta/sá/sú when modifying a noun:
- þessi bók (this book), þetta mál (this issue), sá maður (that man).

How would I say “We all want to solve this together,” and how does the adjective agree?

Við viljum öll leysa þetta saman.
Agreement of “all” with a mixed/unspecified group is neuter plural öll.
- All men (masc.): allir
- All women (fem.): allar
- Mixed/we in general: öll

What are basic principal parts of leysa for other tenses?