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Math Help - Generalised Quaternion Algebra over K - Dauns Section 1-5 no 18

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    Super Member Bernhard's Avatar
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    Generalised Quaternion Algebra over K - Dauns Section 1-5 no 18

    In Dauns book "Modules and Rings", Exercise 18 in Section 1-5 reads as follows: (see attachment)

    Let K be any ring with 1K whose center is a field and  0 \ne x, 0 \ne y \in center K are any elements.

    Let I, J, and IJ be symbols not in K.

    Form the set K[I, J] = K + KI + KJ + KIJ of all K linear combinations of {1, I, J, IJ}.

    Prove that the subring K[I] = K + KI has an automorphism of order 2 defined by  a = a_1 + a_2I \rightarrow \overline{a}  = a_1 - a_2I  for  a_1, a_2 \in K which extends to an inner automorphism of order two of all of K[I,J], where  q \rightarrow  \overline{q} = J_{-1}qJ = (1/y)JqJ for q = a + bJ, a, b  \in K[I] . Show that  \overline{q} = \overline{a} +  \overline{b}J .

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    I can show that K[I] = K + KI is an automorphism, but I am unsure of the meaning of an automorphism having order two - let alone proving it! So I would appreciate some help on showing that the automorphism has order two.

    I cannot however show that it 'extends' to the inner automorphism of order two that is then defined.

    I would appreciate help with these parts of the exercise.

    Peter

    [This has also been posted on MHB]
    Last edited by Bernhard; August 15th 2013 at 04:50 PM.
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    Super Member Bernhard's Avatar
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    Re: Generalised Quaternion Algebra over K - Dauns Section 1-5 no 18

    OK, I have just been checking the order of an automorphism and it is actually quite straightforward - definition is as follows:

    If the automorphism is  f: a \rightarrow \overline{a} as in the exercise that we are focused on, then the order is the smallest  n \ge 1 such that  f^n = E where E is the identity function.

    OK so in the exercise we have that  f: a \rightarrow \overline{a} is obviously of order 2.

    However I am still unable to show that K[I] = K + KI 'extends' (?) to the inner automorphism specified.

    I would very much appreciate some help.

    Peter

    [This has also been posted on MHB]
    Last edited by Bernhard; August 15th 2013 at 04:51 PM.
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