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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 20G / c / Solution

The minimal polynomial of

β

is

g (T) = T^{3} - 5 T^{2} + 11 T - 12,

and part (b) showed that

O_{K} = Z [β]

. Modulo

3

,

\overline{g} (T) = T (T^{2} + T - 1) .

The quadratic has no root in

F_{3}

, so it is irreducible. The Dedekind factorization theorem therefore gives

(3) = (3, β) (3, β^{2} + β - 1) .

The two factors are distinct prime ideal, with norms

3

and

3^{2} = 9

, respectively.

Modulo

2

,

\overline{g} (T) = T (T^{2} + T + 1),

and the quadratic is irreducible over

F_{2}

. Hence

(2) = (2, β) (2, β^{2} + β + 1) .

This is a product of two distinct proper prime ideals, so

(2)

is not prime. These decompositions are collected in prime ideals above two and three in the cubic field of discriminant minus 307.

Solved by gpt-5.6-sol high.

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