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Math Help - Field Extensions - Basic Theory

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    Super Member Bernhard's Avatar
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    Field Extensions - Basic Theory

    I am reading Dummit and Foote Ch 13 on Field Theory.

    On page 515-516 D&F give a series of basic examples on field extensions - see attachment.

    The start to Example (4) reads as follows: (see attachment)

    (4) Let  F = \mathbb{Q} and  p(x) = x^3 - 2 , irreducible by Eisenstein. (by Eisenstein???)

    Denoting a root of p(x) by  \theta we obtain the field

     \mathbb{Q}[x]/(x^3 -2) \cong {a + b \theta + c {\theta}^2 | a,b,c \in \mathbb{Q}

    with  {\theta}^3 = 2 , an extension of degree 3.

    To find the inverse of, say,  1 + \theta in this field, we can proceed as follows:

    By the Euclidean Algorithm in  \mathbb{Q}[x] there are polynomials a(x) and b(x) with

     a(x)(1 + x) + b(x)(x^3 - 2) = 1

    ... ... etc etc

    -----------------------------------------------------------------------------------------

    My problem is this:

    How do D&F get the equation  a(x)(1 + x) + b(x)(x^3 - 2) = 1 ?

    It looks a bit like they are implying that there is a GCD of 1 between (1 + x) and  (x^3 - 2) and then use Theorem 4 on page 275 (see attached) relating the Euclidean Algorithm and the GCD of two elements of a Euclidean Domain, but I am not sure and further, not sure why the GCD is 1 anyway.

    Can someone please clarify the above for me?

    Peter
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