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Math Help - Polynomial rings over Fields

  1. #1
    Super Member Bernhard's Avatar
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    Polynomial rings over Fields

    I am reading Dummit and Foote Section 9.2: Polynomial Rings Over Fields I

    I am having some trouble understanding Example 3 on page 300 (see attached)

    My problem is mainly with understanding the notation and terminology.

    The start of Example 3 reads as follows.

    "If p is a prime, the ring Z/pZ[x] obtained by reducing Z[x] modulo the prime ideal (p) is a Principal Ideal Domain, since the coefficients lie in the field Z/pZ ... ... "

    To me the ring Z/pZ[x] would be formed by reducing Z modulo p to form three cosets, namely  \overline{0}, \overline{1}, \overline{2} and then forming Z/pZ[x] by taking coeffiients from Z/pZ

    I am really unsure what D&F mean by "reducing Z[x] modulo the prime ideal (p)" unless they mean reducing the coefficients of Z[x] to coefficients from Z/pZ.

    I am also assuming that when D&F use the notation Z/pZ[x] they are meaning (Z/pZ)[x]


    Can someone clarify this for me?

    Peter
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  2. #2
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    Re: Polynomial rings over Fields

    Quote Originally Posted by Bernhard View Post
    I am really unsure what D&F mean by "reducing Z[x] modulo the prime ideal (p)" unless they mean reducing the coefficients of Z[x] to coefficients from Z/pZ.
    Yes.
    I am also assuming that when D&F use the notation Z/pZ[x] they are meaning (Z/pZ)[x]
    Yes.

    To me the ring Z/pZ[x] would be formed by reducing Z modulo p to form three cosets, namely  \overline{0}, \overline{1}, \overline{2} and then forming Z/pZ[x] by taking coeffiients from Z/pZ
    This is also valid. The two constructions are isomorphic.
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