Cyclotomic Integers: σg(α) notation

In today's blog, I will review the σg(α) notation from Harold Edward's Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory. I am continuing down the same chapter that I started in an earlier blog. If you would like to start at the beginning of cyclotomic integer properties, start here.

I will use this notation later on when I go over Kummer's proof of Fermat's Last Theorem for regular primes.

Edwards offers the following notation in Section 4.5:

Definition: σg(α)

Let σ signify the conjugate α^γ for a given cyclotomic integer where γ is a primitive root mod λ and λ is a prime integer and α is a primitive root of unity where α^λ = 1.

Example 1: σg(α), σ²g(α)

For a given cyclotomic integer g(α), σg(α) = g(α^γ) and σ²g(α) = g(α^γγ)

Example 2: Ng(α) in terms of σg(α) [See here for more information on Ng(α) notation]

Using this new notation, since γ is a primitive root mod λ, we see that:

Ng(α) = g(α)*σg(α)*σ²g(α)*...*σ^λ-2g(α)

Example 3: σ^λ-1g(α)

Since σ is defined as α → α^γ, and since the order of a primitive root is λ - 1 (see Theorem 1 here), we see that:

σ^λ-1 = g(α^(λ-1)*(γ)) = g(α¹) = g(α)

Note:

The advantage of this notation is that it avoids many of the subscripts that we have been using previously.

Based on the properties of primitive roots (see here), we can see that σg(α), σ²g(α), ..., σ^λ-1g(α) are all distinct.