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I Field Extensions - Remarks by Lovett - Page 326 ... ...

  1. May 8, 2017 #1
    I am reading "Abstract Algebra: Structures and Applications" by Stephen Lovett ...

    I am currently focused on Chapter 7: Field Extensions ... ...

    I need help with some remarks of Lovett following Theorem 7.1.12 and Example 7.1.13 on page 326 ...


    The remarks by Lovett read as follows:



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    In the above remarks from Lovett, we read the following:

    " ... ... In the quotient ring ##K##, this implies that ##\overline{ a(x) q(x) } = 1##. Thus in ##K, \ a( \alpha ) q( \alpha ) = 1##. ... ... "


    My question is as follows:

    Can someone please explain exactly why/how it is that ##\overline{ a(x) q(x) } = 1## implies that ##a( \alpha ) q( \alpha ) = 1## ... ... ?


    Help will be appreciated ...

    Peter
     

    Attached Files:

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  3. May 8, 2017 #2

    andrewkirk

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    In the equation ##a(x)q(x)+b(x)p(x)=1##, substitute ##\alpha## for ##x##. Since ##p(\alpha)=0## (we were told ##\alpha## is a root of ##p(x)##) the equation collapses to the desired result.
     
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