TGS Announces 2023 Results and Strategic Plan Update

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Proposition 1.12.

Let c ≥ 1. ε > 0 is an algebraic number in Qp(ℓ), where p > 1 is a prime and ℓ ≥ 1 is an integer. Then ε is contained in the union of finitely many algebraic intervals with endpoints which are p-th roots of 1, for 1 ≤ j ≤ ε c .

Proof.

For ε ∈ Qp(ℓ) and 1 ≤ j ≤ ε c , set and

Note that for 1 ≤ j ≤ ε c . For ε ∈ Qp(ℓ) and 1 ≤ j ≤ ε c , we have

(1 + ε) j ≤ (1 + ε) j = (1 + ε) j ≤ (1 + ε) j.

Now let ε ∈ Qp(ℓ) and ε c ≤ j ≤ n. Then and

Hence, we have

which implies that

Hence, we

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