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Math Help - prime

  1. #1
    Senior Member Sampras's Avatar
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    prime

    Show that the principal ideal  (x-1) in  \mathbb{Z}[x] is prime but not maximal.

    To show it is prime consider  ab \in (x-1) . Then we want to show that  a \in (x-1) or  b \in (x-1) .

    Now  (x-1) = \{c(x-1): c \in \mathbb{Z}[x] \} . So  ab = c[x-1] for some polynomial  c \in \mathbb{Z}[x] . This implies that either  a \in (x-1) or  b \in (x-1) since we can take the other to be the unit polynomial.


    Suppose for contradiction that it was maximal. Then if  (x-1) \subseteq J \subseteq \mathbb{Z}[x] , either  (x-1) = J or  J = \mathbb{Z}[x] . Consider  (2x) . Then it doesn't contain  x^2+1 = x(x+1) . Thus  (x+1) is not maximal.

    Is this correct?
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  2. #2
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    Quote Originally Posted by Sampras View Post
    Show that the principal ideal  (x-1) in  \mathbb{Z}[x] is prime but not maximal.

    To show it is prime consider  ab \in (x-1) . Then we want to show that  a \in (x-1) or  b \in (x-1) .

    Now  (x-1) = \{c(x-1): c \in \mathbb{Z}[x] \} . So  ab = c[x-1] for some polynomial  c \in \mathbb{Z}[x] . This implies that either  a \in (x-1) or  b \in (x-1) since we can take the other to be the unit polynomial.


    Uuh?? What "other"? And what is "the unit polynomial"? Perhaps this is simpler: a polynomial is divisible by (x-1) iff 1 is one of its roots (this is the residue theorem for (long) division of polynomials), so ab\in (x-1) \Longleftrightarrow ab(1)=0 \Longleftrightarrow a(1)=0 \;or \;b(1)=0...

    Suppose for contradiction that it was maximal. Then if  (x-1) \subseteq J \subseteq \mathbb{Z}[x] , either  (x-1) = J or  J = \mathbb{Z}[x] . Consider  (2x) . Then it doesn't contain  x^2+1 = x(x+1) . Thus  (x+1) is not maximal.

    Is this correct?
    Please do read now, coldly, what you wrote: can you understand it? Because if you can then you're not explaining it clearly at all, and from things like this marks in exams go away for vacation in Acapulco:

    (x-1)\subset (2,x-1)

    Tonio
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  3. #3
    Senior Member Sampras's Avatar
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    What does the notation  (2,x-1) mean?
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    Quote Originally Posted by Sampras View Post
    What does the notation  (2,x-1) mean?

    Err...the ideal generated by x-1 and 2, also denoted by 2\mathbb{Z}[x] +(x-1)\mathbb{Z}[x]

    Tonio
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