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I Galois Theory - Fixed Subfield of K by H ...

  1. Jun 16, 2017 #1
    I am reading "Abstract Algebra: Structures and Applications" by Stephen Lovett ...

    I am currently focused on Chapter 8: Galois Theory, Section 1: Automorphisms of Field Extensions ... ...

    I need help with Proposition 11.1.11 on page 560 ... ...


    Proposition 11.1.11 reads as follows:


    ?temp_hash=b73d2c3295f7984285d98c67b26cdd93.png



    In the above Proposition from Lovett we read the following:


    " ... ... Since ##\sigma## is a homomorphism ##U(F) \ \rightarrow \ U(F)## ... ... "


    My question is ... ... what is ##F## ... is it a typo ... should it be ##K ##...


    Hoping someone can help ... ...

    Peter


    NOTE: ##U(F)## in Lovett means the group of units of the ring ##F## ...
     

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  3. Jun 16, 2017 #2

    andrewkirk

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    It looks like a typo to me. ##F## is first mentioned at the end of the second line, in ##(F,+)##. If we replace ##F## by ##K## the proof still works.

    Is it possible that, in the context of the book, ##K## has been assumed to be a sub-field of some larger field ##F##? If so, and that has been stated in the preceding pages, then it wouldn't be a typo.

    It's also not clear to me what ##U(F)## means, although from the context I'm guessing it's referring to the multiplicative group ##(F-\{0\},\times)## which, if we assume that ##F## really means ##K##, is ##(K-\{0\},\times)##.
     
  4. Jun 16, 2017 #3

    fresh_42

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    FWIW. I recall a situation, in which ##F \subseteq K## has been the extension. I remember it, as I confused both and finally built a mnemonic ##F =## fixed elements.
     
  5. Jun 16, 2017 #4
    Hmm ... yes ... Lovett often uses K/F for a field extension ... but still ... seems to me that that doesn't resolve the problem of the exact nature of F ...

    Do you think that Lovett meant K when he wrote F?

    Peter
     
  6. Jun 16, 2017 #5

    fresh_42

    Staff: Mentor

    I have the same difficulties as @andrewkirk to understand what ##F## and ##U(F)## are. ##U(.)## could be units, the multiplicative group of a field, and ##F=K## which makes sense, if I didn't miss something. As the entire topic is about the correspondence between fields and automorhism groups, it might well be that a ##F## somehow found its way into the proof although it should have been ##K##.
     
  7. Jun 16, 2017 #6
    HI fresh_42 ...

    The notation U(K) in Lovett is the group of units of K ...

    Thanks ... I also think F should be K ... glad to have your agreement ...

    Peter
     
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