There are no 3×3 matrices over QQ of multiplicative order 8

Show that there do not exist 3 \times 3 matrices A over \mathbb{Q} such that A^8 = I and A^4 \neq I.


If A is such a matrix, then the minimal polynomial of A divides x^8-1 = (x^4-1)(x^4+1) but not x^4-1; so the minimal polynomial of A divides x^4+1. Since A has dimension 3, the characteristic polynomial of A has degree 3, so the minimal polynomial has degree at most 3. Note that (x+1)^4+1 = x^4 + 4x^3 + 6x^2 + 4x + 2 is Eisenstein at 2, and so x^4+1 is irreducible over \mathbb{Q}. (See Proposition 13 on page 309 in D&F, and Example 3 on page 310.) So we have a contradiction- no divisor of x^4+1 over \mathbb{Q} can have degree between 1 and 3. So no such matrix A exists.

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