Can anyone help me with the following exercise from Dummit and Foote?(adsbygoogle = window.adsbygoogle || []).push({});

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Describe the centre of the real Hamilton Quaternions H.

Prove that {a + bi | a,b R} is a subring of H which is a field but is not contained in the centre of H.

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Regarding the problem of describing the centre - thoughts so far are as follows:

Let h = a + bi + cj + dk

Then investigate conditions for i and h to commute!

i [itex] \star [/itex] h = i [itex] \star [/itex] ( a + bi + cj + dk)

= ai + b[itex]i^2[/itex] + cij + dik

= ai - b + ck - dj

h [itex] \star [/itex] i = ( a + bi + cj + dk) [itex] \star [/itex] i

= ai + b[itex]i^2[/itex] + cji + dki

= ai - b -ck + dj

Thus i and h commute only if c = d = 0

Proceeding similarly we find that

j and h commute only if b = d = 0

and

k and h commute only if b = c = 0

Thus it seems as if I am being driven to the conclusion that the only Hamilton Quaternions that commute with every element of the ring of Hamilton Quaternions are elements of the form

a + 0i + 0j + 0k

But I am unsure of how to formally and validly argue from the facts established above to conclude this!

Can anyone help or at least confirm that I am on the right track!

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# Centre of the real Hamilton Quaternions H

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