 Solution

ID: past-exam-of-the-algebra-and-analysis-ii-for-mathematics-and-philosophy-course-of-the-university-of-oxford/2013/1/b/solution

F [x]

f = q m_{T} + r, r = 0 or de g r < de g m_{T} .

T

gives

r (T) = f (T) - q (T) m_{T} (T) = 0

. A nonzero remainder could be rescaled to a monic polynomial annihilating

T

of smaller degree, contradicting the defining minimality. Hence

r = 0

and

m_{T} ∣ f .

In particular, the annihilating polynomials form the principal ideal

(m_{T})

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