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Math Help - Polynomial Rings - Irreducibility - Proof of Eisenstein's Criteria

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    Super Member Bernhard's Avatar
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    Polynomial Rings - Irreducibility - Proof of Eisenstein's Criteria

    I am reading Dummit and Foote Section 9.4 Irreducibility Criteria. In particular I am struggling to follow the proof of Eisenstein's Criteria (pages 309-310 - see attached).

    Eisenstein's Criterion is stated in Dummit and Foote as follows: (see attachment)

    Proposition 13 (Eisenstein's Criterion) Let P be a prime ideal of the integral domain R and let

     f(x) = x^n + a_{n-1}x^{n-1} + ... ... + a_1x + a_0

    be a polynomial in R[x] (here  n \ge 1 )

    Suppose  a_{n-1}, ... ... a_1, a_0 are all elements of P and suppose  a_0 is not an element of  P^2 .

    Then f(x) is irreducible in R[x]


    The proof begins as follows: (see attachment)

    Proof: Suppose f(x) were reducible, say f(x) = a(x)b(x) in R[x] where a(x) and b(x) are nonconstant polynomials.

    Reducing the equation modulo P and using the assumptions on the coefficients of f(x) we obtain the equation  x^n = \overline{a(x)b(x)} in (R/P)[x] where the bar denotes polynomials with coefficients reduced modulo P. Since P is a prime ideal, R/P is an integral domain, and it follows that  \overline{a(x)} and  \overline{b(x)} have zero constant term i.e. the constant terms of both a(x) and b(x) are elements of P. But then the constant term a_0 of f(x) as the product of these two would be an element of  P^2 , a contradiction.


    Questions/Issues

    (1) I cannot see exactly how the following follows:

    "Reducing the equation modulo P and using the assumptions on the coefficients of f(x) we obtain the equation  x^n =  \overline{a(x)b(x)} in (R/P)[x]"

    I am struggling to see just exactly how this follows: can someone please clarify this?

    (2) I am somewhat unsure of the following:

    "t follows that  \overline{a(x)} and  \overline{b(x)} have zero constant term"

    This probably is a consequence of the mechanics of the reduction modulo P process but can someone please clarify exactly how it follows?

    Peter
    Last edited by Bernhard; May 24th 2013 at 08:38 PM.
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