 Solution

ID: past-exam-of-the-rings-and-modules-course-of-the-university-of-oxford/2023/3/b/iv/solution

Each entry of the matrix

Q

from part (b)(iii) has unique real and imaginary integer coefficients. Write

Q = U + iV, U, V \in M_{2} (Z) .

It already belongs to the general linear group

GL_{2} (Z [i])

. Expanding the relation from part (b)(iii) gives

0 = (i I - A) (U + iV) e_{2} = (- A U - V) e_{2} + i (U - A V) e_{2} .

The real and imaginary parts must both vanish, so

A U e_{2} = - V e_{2}, A V e_{2} = U e_{2}, U + iV \in GL_{2} (Z [i]) .

These are the required identities, with

e_{2} = (0, 1)^{T}

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