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I’m learning about “Finalsätze” and I got the following example from my teacher.
Ich lerne fleißig Deutsch. Mein Chef soll zufrieden sein.
Ich lerne fleißig Deutsch, damit mein Chef zufrieden ist.
Why isn’t it meinen Chef? I interpret that Chef is in accusative?
Do you already understand why "Mein Chef" in the sentence "Mein Chef soll zufrieden sein." is nominative and not accusative?
If no: Hubert Schölnast's answer is what you are looking for.
If your problem is only "mein Chef" in "damit mein Chef zufrieden ist." the following question from your comment is the key:
I don't know what "subordinate clause" is?
When I understood your profile correctly you speak Swedish and that language also seems to have subordinate clauses. When combining two sentences to one sentence one of the sentences remains a main clause while the other sentence becomes a subordinate clause.
The same is true for English language:
Sentence 1: I want to get the job.
Sentence 2: Therefore I learn German.
Combined sentence: I learn German because I want to get the job.
In English language the word order in subordinate sentences does not change; in German language the word order of subordinate clauses differs from the word order in main clause:
Main clause: Ich will die Stelle bekommen.
Subordinate clause: ... ich die Stelle bekommen will.
... however the elements of a subordinate clause (Subject, dative object and accusative object) occuring in a subordinate clause are the same as in the main clause. Example:
Sentence 1: Mein Chef soll meiner Schwester eine Stelle geben.
Sentence 2: Sie lernt fleißig Deutsch.
Combined: Weil mein Chef meiner Schwester eine Stelle geben soll, lernt sie fleißig Deutsch.
In this example "Sentence 1" becomes a subordinate clause while "Sentence 2" reamains a main clause. "Sentence 1" has a subject ("mein Chef"), a dative object ("meiner Schwester") and an accusative object ("eine Stelle"). For this reason the subordinate clause in the combined sentence also has these elements.
The sentence from your example ("Mein Chef soll zufrieden sein.") contains a subject (nominative) and no objects (dative, accusative). Therefore the subordinate clause also only contains a subject.
You have two sentences. Either as two distinct sentences (in line 1), or joined together to one longer sentence via the subjunctive conjunction »damit« (in line 2).
In German a full sentence needs at least two parts of speech (same in English and almost all other languages spoken by humans):
In German the subject is ALLWAYS in nominative case. No Exception of this rule.
In your example the action is »to be happy« (zufrieden sein). This is the verb.
The part of speech that is happy (or has to be happy), i.e. that part of speech that performs this action is »my boss« (mein Chef). So this is the subject, and because it is the subject, it has to be in nominative case. And the preposition mein is part of the subject and therefore also has to stand in nominative case.
