 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-2/18g/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

Let

σ (α) = ξ_{d} α

with

α \neq = 0

. Then

σ^{j} (α) = ξ_{d}^{j} α (0 \leq j < d),

and these

d

elements are distinct because

ξ_{d}

is a primitive root of unity. Thus the stabilizer of

α

in the cyclic Galois group is trivial. The orbit-stabilizer form of the Fundamental theorem of Galois theory gives

[K (α) : K] = d = [L : K],

L = K (α) .

Moreover,

σ (α^{d}) = ξ_{d}^{d} α^{d} = α^{d} .

K

, and therefore

α^{d} \in K .

This is the basic eigenvector mechanism behind a cyclic Kummer extension.

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