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    Explain why \mathbb{Z}[x] is a UFD (Unique Factorisation Domain). Assume nothing.
    Can I assume that \mathbb{Z}[x] is a ring (or even an ID (Integral Domain) for that matter)? I ask of the 'assume nothing' within the question.
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    (Original post by RamocitoMorales)
    Can I assume that \mathbb{Z}[x] is a ring (or even an ID (Integral Domain) for that matter)? I ask of the 'assume nothing' within the question.
    You can definitely assume that it is a ring. I mean, if you read \mathbb{Z}[x] aloud, you would say "the ring of polynomials with integer coefficients".

    Also, being a UFD is a property of rings, so from the point of view of the question making sense you could say that \mathbb{Z}[x] being a ring is tacitly assumed.

    On the other hand, I am not sure how you are meant to answer the question aside from that. I would have thought 'assume nothing' would mean that you can't assume that it is a domain or use, for instance the fact that a polynomial extension of a UFD is a UFD.

    My best guess would be to say that you should be able to assume the fund. theorem of arithmetic (since this presumably comes before any algebra course) and that you should mimic the proof of the fact that the polynomial extension of a UFD is a UFD in the special case of the integers.
 
 
 
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