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

• Aug 15th 2013, 05:44 PM
Bernhard
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]
• Aug 15th 2013, 05:48 PM
Bernhard
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]