 Solution

ID: past-exam-of-the-number-theory-course-of-the-university-of-oxford/2011/b/6/c/solution

For degree

3

there are either three real embeddings or one real embedding and one complex pair. The respective Minkowski bound for ideal classes are

\frac{2}{9} ∣ Δ^{2} (L) ∣, \frac{8}{9 π} ∣ Δ^{2} (L) ∣ .

Under the given discriminant bound they are at most

\frac{2}{9} 45 < 2

and

\frac{8}{9 π} 45 < 2

. Hence every ideal class contains an integral ideal with positive integer norm strictly below

2

, so with norm

1

. The only such ideal is

O_{L}

. Therefore

h_{L} = 1.

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