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For example, if I compose a chord progression F-G-C-Am.
Can I Assume C as the Key? How do I know I'm not in A minor?
I've seen songs analysis that says a song is in determined key even when the root is not played once in the whole song, how does this works? I understand that by listening how the song resolves I can know the key, but its there any rule?
Saying that a piece of music is "in a key" often (even usually?) isn't a statement of fact, but a statement of perspective.
When you say that a piece is in C major, you're saying that
The thing about both those statements is that they leave some room for opinion - they are not absolute, factual statements.
I compose a chord progression F-G-C-Am … can I assume C as the Key?
Not necessarily. It depends on the phrasing of those chords. If Am is the most important chord in the phrasing, then it might sounds more to you like A minor. You have to think whether you feel that C, or A, or something else is the 'home note'.
how do I know I'm not in Am
You can't "know" - but you can feel that the piece doesn't come home to A.
I've seen songs analysis that says a song is in determined key even when the root is not played once in the whole song, how does this works?
Because key is a question of perspective, not a question of fact, you can analyse any song from the point of view of any key. It might be useful and interesting to analyse a given piece from the point of view of a key for which the song didn't land on the tonic chord. In such cases, it's unlikely that that would be the only sensible key to think about.
is there any rule?
No, no absolute rule. People often come up with "rules of thumb" that work for them, but none of them are definitive.
The one way in which being "in a key" could be definitive is if the composer labels it as such - e.g. if I write "Scherzo and Trio in A minor", then in that sense it is defined to be "the key" of the piece. But I don't think that's the sense in which you ask the question.
One shows a key by the main cadences in the piece; especially at the end of the whole composition. Some internal cadences (at the end of phrases or sections matter too.)
If a piece is in C-major, the piece will probably end with an authentic cadence on a C-maor chord; usually the pattern G-C or G7-C or dm-G-C or dm-G-G or F-G7-G something similar (ii-V-I or IV-V-I).
The cadence generally has a pre-dominant chord (or subdominant, depending on who's doing the analysis), a dominant chord (maybe with a 7th) followed by the tonic chord. Symbolically one has S-D-T (subdominant, dominant, tonic). S stands for things like ii, ii0, ii6, IV; D stands for V or V7, and T for I in a piece-ending cadence.
In a minor key, the same pattern occurs, S-D-T where S now may be iv or ii0 or ii06 etc, and D is V or V7 (just as in the parallel major) and T as i.
So if the piece ends with a G-C chord pattern perhaps preceded by a dm or dm6 or F major, the piece (or at least the previous section) is in C-major.
If the piece ends with an E-a or d-E-a or f#0-E7-a, etc. the piece is in A-minor.
Pieces do change key but the sections (usually are clearly marked).
