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it would help if you could share where you found the application link. I'd expect you need to at least be able to speak Icelandic but I can't say I've ever met anyone who works in the jail!

Found it on their website here:

fangelsi.is/fangelsismalastofnun-rikisins/fangavardaskoli/

do you have a kennitala? The registration process starts with entering your kennitala (Icelandic ID number). It does seem to be checking against the national registry because just typing in a bunch of numbers does not work.

Other than that, it depends on what you're applying for- the guard application wants Icelandic abilities, but the "specialist" doesn't state any specific requirements since it depends on what kind of specialist you are.

the link you provided there is for a jail school. I guess it teaches you how to be a good, professional jailer, so the application form on that link is for applying to the school. Here is where the job information is:

Do you have a visa to work in Iceland (EU)?
If not, you won't be able to apply for any jobs in Iceland.

You won’t be able to fill inn any online application forms without “kennitala” (Social ID)
I'm not sure Icelandic is a requirement. You can always e-mail them, ask them and apply that way (regarding you have a visa). You could give them the social ID number later.