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I Basics of Field Extensions ... ... Ireland and Rosen, Ch 12

  1. Mar 30, 2017 #1
    I am reading Kenneth Ireland and Michael Rosen's book, "A Classical Introduction to Modern Number Theory" ... ...

    I am currently focused on Chapter 12: Algebraic Number Theory ... ...

    I need some help in order to follow a basic result in Section 1: Algebraic Preliminaries ...

    The start of Section 1 reads as follows:



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    QUESTION 1


    In the above text by Ireland and Rosen, we read the following:


    "... ... Suppose ##\alpha_1, \alpha_2, \ ... \ ... \ , \alpha_n## is a basis for ##L/K## and ##\alpha \in L##.

    Then ##\alpha \alpha_i = \sum_j a_{ ij } \alpha_j## with ##a_{ ij } \in K## ... ... "


    "My question is ... ... how do Ireland and Rosen get ##\alpha \alpha_i = \sum_j a_{ ij } \alpha_j ## ... ... ?



    My thoughts are as follows ...


    Given ##L/K##, we have that ##L## is a vector space over ##K##.

    ... we then let ##\alpha_1, \alpha_2, \ ... \ ... \ , \alpha_n## be a basis for ##L## as a vector space over ##K##

    ( i take it that that is what I&R mean by "... ... Suppose ##\alpha_1, \alpha_2, \ ... \ ... \ , \alpha_n## is a basis for ##L/K##")


    ... we then let ##\alpha \in L## ... ... then there exist ##a_1, a_2, \ ... \ ... \ , a_n \in K##


    such that


    ##\alpha = a_1 \alpha_1 + a_2 \alpha_2 + \ ... \ ... \ a_n \alpha_n##


    so that


    ##\alpha \alpha_i = ( a_1 \alpha_1 + a_2 \alpha_2 + \ ... \ ... \ a_n \alpha_n ) \alpha_i ## ... ... ... (1)


    ... BUT ...

    Ireland and Rosen write (see above)


    ##\alpha \alpha_i = \sum_j a_{ ij } \alpha_j##


    ##= a_{ i1 } \alpha_1 + a_{ i2 } \alpha_2 + \ ... \ ... \ + a_{ in } \alpha_n## ... ... ... (2)


    My question is ... how do we get expression (1) equal to (2) ... ...




    QUESTION 2


    In the above text by Ireland and Rosen, we read the following:


    "... ...The norm of ##\alpha, N_{ L/K } ( \alpha )## is ##\text{ det} (a_{ ij }) ## ... ...


    I cannot fully understand the process involved in forming the norm ... can someone please explain ... preferably via a simple example ...



    Hope someone can help ...


    Peter
     

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    Last edited: Mar 30, 2017
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  3. Mar 30, 2017 #2

    andrewkirk

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    In Question 1, we observe that, since ##\alpha_1,...,\alpha_n## form a basis for ##L## as a vector space over ##K##, any element of ##L## can be written as a linear combination of those basis elements, with coefficients in ##K##, that is, as ##\sum_{j=1}^na_j\alpha_j## with ##a_j\in K\forall j##.

    Since ##\alpha,\alpha_i## are both in ##L##, which is a field, ##\alpha\alpha_i## must also be in ##L## and hence can be written as such a linear sum. We then just relabel each ##a_j## as ##a_{ij}## and we have the text's formula.

    In question 2, note that, given ##\alpha\in L## and a basis ##\alpha_1,...,\alpha_n## for ##L##, each ##\alpha_i## gives us a set of ##n ## coefficients in ##K##: ##a_{i1},....,a_{in}##. Since there are ##n## ##\alpha_i##s, we can put those coefficients in a ##n\times n## matrix and then calculate a determinant of that matrix.
     
  4. Mar 31, 2017 #3
    Thanks Andrew ... just reflecting on what you have written ...

    Peter
     
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