Learn Hardcore Icelandic: Lögreglan setur skilti við veginn þegar umferðin er hættuleg. - The police put a sign by the road when the traffic is dangerous.

lögregla is a feminine noun.
The definite article for feminine nouns in the nominative singular is -in, but it often merges with the final vowel:
- lögregla + in → lögreglan (the a and i merge and give -an in spelling/pronunciation).

So lögreglan literally means the police, and that’s why there’s no separate word like the in front of it.

What exactly is setur, and why is that form used here instead of setja?

Setur is the 3rd person singular present tense of the verb setja (to put / place / set).

Infinitive: að setja = to put / to place
1st person singular: ég set
2nd person singular: þú setur
3rd person singular: hann / hún / það setur
1st person plural: við setjum
etc.

In the sentence, the subject is lögreglan (the police), which is grammatically 3rd person singular, so you need the form:

Lögreglan setur … = The police put / set …

The present tense in Icelandic is often used, just like in English, to describe general rules or habitual actions, which fits this sentence.

What case is veginn in, and what does the preposition við do to it?

Veginn is in the accusative case, definite singular.

The base noun is vegur = road (masculine).
Accusative singular indefinite: veg
Accusative singular definite: veginn

The preposition við (by, at, near, against) always takes the accusative case. So:

við + vegur → við veginn = by the road

So the structure is:

við (preposition that takes accusative)
veginn (accusative singular definite of vegur)

Why is it við veginn and not á veginum? Are both possible?

Both can be possible, but they mean different things:

við veginn = by / near / at the side of the road
- við suggests beside or at something.
á veginum = on the road / on the surface of the road
- á means on or in depending on context and takes dative here, so veginum is dative singular definite.

In your sentence:

Lögreglan setur skilti við veginn …
Means the police place a sign by the roadside, not literally on the surface of the road.

If you said:

Lögreglan setur skilti á veginum …
it would sound more like they are placing the sign on the road itself (for example, standing on the road), which is a different mental picture.

Is skilti singular or plural here, and why doesn’t it look different in the plural?

Skilti is a neuter noun where the nominative and accusative forms of singular and plural look the same.

Singular nominative: skilti
Singular accusative: skilti
Plural nominative: skilti
Plural accusative: skilti

So in this sentence skilti could mean a sign or signs; the form alone is ambiguous. Context usually decides how you translate it.

The definite forms are different:

Singular definite: skiltið = the sign
Plural definite: skiltin = the signs

Since there is no -ð or -n here, it’s indefinite, and could be translated as a sign or signs depending on what you want to say in English.

Why does umferðin have -in at the end, and is traffic always singular in Icelandic?

Umferð is a feminine noun meaning traffic.

umferð = traffic (indefinite)
umferðin = the traffic (definite)

The -in at the end is again the definite article for feminine nouns in the nominative singular:

umferð + in → umferðin = the traffic

Like English, Icelandic treats traffic / umferð as a mass noun, so it is normally singular. You don’t say umferðir to mean different cars in traffic; umferðin already covers the traffic (as a whole).

Why is the adjective hættuleg in that form, and what is it agreeing with?

Hættuleg means dangerous and is an adjective. Icelandic adjectives must agree with the noun they describe in:

Gender (masculine / feminine / neuter)
Number (singular / plural)
Case (nominative / accusative / dative / genitive)

Here, hættuleg describes umferðin (the traffic):

umferðin is feminine, singular, nominative (subject of its clause).
Therefore the adjective must be feminine singular nominative: hættuleg.

If the noun were masculine or neuter, you would see different endings, for example:

bíllinn er hættulegur = the car is dangerous (masc. sg. nom.)
veðrið er hættulegt = the weather is dangerous (neut. sg. nom.)

Why do we say umferðin er hættuleg and not just hættuleg umferðin?

In this sentence, hættuleg is a predicative adjective (used after to be), not an adjective directly attached in front of the noun.

Structure:

umferðin = the subject (the traffic)
er = the verb is
hættuleg = the predicate describing the subject (dangerous)

So the pattern is:

[Noun] + er + [adjective]
- umferðin er hættuleg = the traffic is dangerous

If you put the adjective directly in front of the noun, you get an attributive use:

hættuleg umferð = dangerous traffic
hættuleg umferðin = the dangerous traffic

Both are correct Icelandic, but they mean different things and fit into sentences differently. The original sentence wants a full clause (when the traffic is dangerous), so it uses er.

What does þegar do in this sentence, and can I move the þegar-clause to the front?

Þegar is a subordinating conjunction meaning when (in the temporal sense).

In your sentence:

þegar umferðin er hættuleg = when the traffic is dangerous

It introduces a subordinate clause of time. The basic word order in that clause stays subject – verb – complement:

umferðin (subject) er (verb) hættuleg (adjective)

You can absolutely move the þegar-clause to the front, just like in English:

Þegar umferðin er hættuleg, setur lögreglan skilti við veginn.
= When the traffic is dangerous, the police put(s) a sign/signs by the road.

In writing, you normally add a comma when the þegar-clause comes first.

Could this sentence be in the past or future, and how would it change?

Yes. Only the verb setur (and possibly er) needs to change tense.

Present (original):
Lögreglan setur skilti við veginn þegar umferðin er hættuleg.
= The police put / place a sign when the traffic is dangerous. (general rule)
Past:
Lögreglan setti skilti við veginn þegar umferðin var hættuleg.
- setti = past of setja
- var = past of vera (to be)
  = The police put / placed a sign when the traffic was dangerous.
Future (using munu):
Lögreglan mun setja skilti við veginn þegar umferðin er hættuleg.
= The police will put a sign when the traffic is dangerous.

Icelandic often uses the simple present with time adverbs instead of a formal future tense, but munu + infinitive is a straightforward future-like construction.

Could I say Lögreglan setur upp skilti instead of setur skilti? What is the difference?

Yes, you can say:

Lögreglan setur upp skilti við veginn …

Setja upp is a verb + particle combination that typically means to put up / to erect / to install (especially for things like signs, posters, decorations).

Subtle difference:

setja skilti = to place / set a sign (more neutral)
setja upp skilti = to put up a sign (with the nuance of installing/erecting it)

In practice, for signs, setja upp skilti is very common and sounds very natural.

How would I change the sentence if I specifically want the signs (definite plural) instead of just sign(s)?

You need the definite plural of skilti, which is skiltin:

skilti (indefinite, could be singular or plural)
skiltin (definite plural) = the signs

So the sentence becomes:

Lögreglan setur skiltin við veginn þegar umferðin er hættuleg.
= The police put the signs by the road when the traffic is dangerous.

If you wanted to emphasize some signs (indefinite plural) in English, Icelandic would still just use skilti; there is no separate plural ending in the accusative here. Context does the work.

What are the genders of lögregla, skilti, vegur, and umferð, and why does that matter?

lögregla = feminine
skilti = neuter
vegur = masculine
umferð = feminine

It matters because:

The definite endings depend on gender and case:
- lögregla → lögreglan (fem. nom. sg. definite)
- vegur → veginn (masc. acc. sg. definite)
- umferð → umferðin (fem. nom. sg. definite)
Adjectives must agree with the noun:
- umferðin er hættuleg
  - umferðin is feminine singular nominative
  - hættuleg is feminine singular nominative

If umferð were masculine or neuter, the form of hættuleg would change (e.g. hættulegur, hættulegt). Understanding noun gender helps you choose the right endings for both nouns and adjectives.