Lifting the exponent is a popular result in elementary number theory. It basically says that if and , then for an odd prime coprime to and for ,

**Theorem 1. (Lifting the exponent)** .

Here, as usual, means that but .

We shall generalise this to general rings of algebraic integers. In the following, will denote an algebraic number field and its ring of integers. Let such that the ideals are coprime to , and .

**Theorem 2.** .

*Proof.* We proceed by induction on . The base case is clear. Suppose that the assertion holds for . We want to show that , that is,

By hypothesis, . So we need to show that .

Let . We want to show that .

Since , we have . Hence , i.e. . Hence . Let for some . Then

If this is , then , which is impossible by unique factorisation as and are coprime.

**Corollary.** Let be the Fibonacci sequence. Then

(Refer to this post for notation.)

*Proof. *Take (with and ) and let . We proceed by induction on . Suppose for some . By hypothesis, so . Then , so , i.e. . Hence which implies , a contradiction.

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