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Math Help - Factor Rings of Polynomials Over a Field

  1. #1
    Super Member Bernhard's Avatar
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    Factor Rings of Polynomials Over a Field

    On page 222 of Nicholson: Introduction to Abstract Algebra in his section of Factor Rings of Polynomials Over a Field we find Theorem 1 stated as follows: (see attached)

    Theorem 1. Let F be a field and let  A \ne 0 be an ideal of F[x]. Then a uniquely determined monic polynomial h exists exists in F[x] such that A = (h).

    The beginning of the proof reads as follows:

    Proof: Because  A \ne 0 , it contains non-zero polynomials and hence contains monic polynomials (being an ideal) ... ... etc. etc.

    BUT! why must A contain monic polynomials??

    Help with this matter would be appreciated!

    Peter
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  2. #2
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    Re: Factor Rings of Polynomials Over a Field

    Hi,
    Suppose g(x)=a_nx^n+... \text{ is in A with }a_n\ne0.
    Since A is an ideal, a_n^{-1}g(x)} is monic and is in A.
    Thanks from Bernhard
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