Learn Hardcore Icelandic: Bardaginn byrjar úti.

It is capitalized only because it is the first word of the sentence. Icelandic does not capitalize all nouns the way German does. If the same word appeared later in the sentence, it would normally be written bardaginn.

Why is there no separate word for the?

In Icelandic, the definite article is usually attached to the end of the noun instead of being written as a separate word. So bardaginn means the battle.

This is very common in Icelandic:

bardagi = battle
bardaginn = the battle

What is the dictionary form of bardaginn?

The dictionary form is bardagi, which is a masculine noun. Icelandic dictionaries usually list nouns in the indefinite nominative singular form, so you would look up bardagi, not bardaginn.

What case is bardaginn, and why?