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Automorphism Group

  1. Nov 3, 2011 #1
    1. The problem statement, all variables and given/known data

    Let K = Q(21/4)

    Determine the automorphism group Aut(K/Q)


    2. Relevant equations

    An automorphism is an isomorphism from a Field to itself

    Aut(K/Q) is the group of Automorphisms from k/Q to K/Q

    Definition: A K-Homomorphism from L/K to L'/K is a homomorphism L---> L' that is the identity on K

    3. The attempt at a solution

    I am completely at a loss really. I have calculated there are four homomorphisms from K to C and think from there if I know how many are K-homomorphisms then that'll be the number of automorphisms, because a homomorphism from a field to itself is an automorphism (Please correct me if I'm wrong on this). Then that'll give me the set of Automorphisms.

    My problem is that I don't know how to go from the number of homomorphisms to the actual homomorphisms. I think it has a relation to the roots of 2(1 /4) in C (which I have calculated to be 2(1 /4), - 2(1 /4), i*2(1 /4), -i2(1 /4) )

    Please help, this lack of understanding is preventing me from moving forward with other questions and my notes from lectures completely gloss over how to do this.
     
    Last edited: Nov 3, 2011
  2. jcsd
  3. Nov 3, 2011 #2
    one problem that I see is that not every homomorphism is an automorphism. Consider the identity mapping. this is clearly a homomorphism but it is not bijective.
    Also you said there are four homomorphisms from K to C. But what is C? is C just Aut(K/Q)?
    Also I just want to clarify something: Is this what you mean by K:

    [tex] K= \{ a+b \sqrt[4]{2} :a,b \in \mathbb{Q} \} [/tex]
     
    Last edited: Nov 3, 2011
  4. Nov 3, 2011 #3

    Deveno

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    Science Advisor

    no C is not Aut(K/Q).

    Aut(K/Q) is the group of all automorphisms of K that fix Q.

    note K is NOT {a+b(4√2)}, this set is not even closed under multiplication, K is more precisely the set:

    {a + b(4√2) + c√2 + d(4√8)}

    which as a vector space is isomorphic to Q4.

    what IS clear, is that any element of Aut(K/Q) sends 4√2 to another 4th root of two, and that any such automorphism is completely determined by the image of 4√2.

    there are exactly 4 of these in C:

    4√2, -4√2, (4√2)i and -(4√2)i.

    if we agree to set α = 4√2, of the maps:

    α→α
    α→-α
    α→iα
    α→-iα

    how many of these are in Aut(K/Q)?

    (the complex numbers play only an indrect role, here, as a field contaning the algebraic closure of Q(α). we could have just used this field (call it E) instead, but it doesn't lend itself to a nice easy description like C does. the important fact is that there is SOME subfield of C that is the algebraic closure of Q(α), as Q(α) is a subfield of R, and C is the algebraic closure of R (a fact called the fundamental theorem of algebra). note we don't even need all of C, just A, where A is the set of all (real) algebraic numbers).
     
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