 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/ib/paper-4/12f/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

1 + x + x^{3} + x^{4} = (1 + x) (1 + x^{3})

holds over every field. Both factors have positive degree, including in characteristics

2

and

3

, because their leading coefficients remain one. Therefore there is no field over which this polynomial is irreducible.

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