Cyclotomic Integers: Units and Primes

Today's blog continues the discussion of Kummer's proof of Fermat's Last Theorem for regular primes. If you would like to review the historical context for this proof, start here.

Today, I will continue reviewing the basic properties of cyclotomic integers. Today's content comes directly from Chapter 4 of Harold M. Edwards' Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory.

1. Unit

Definition 1: Cyclotomic Unit

A cyclotomic unit is a cyclotomic integer whose norm is 1.

So, if f(α) is a unit, then f(α)*f(α²)*...*f(α^λ-1) = 1.

Definition 2: Cyclotomic Inverse

If f(α) is a unit, then f(α²)*..*f(α^λ-1) is called the inverse and it represented as f(α)^-1.

Lemma 1: if f(α)*g(α) = 1, then f(α) is a unit and g(α) = f(α²)*f(α³)*...*f(α^λ-1).

Proof:

(1) Let f(α)*g(α) = 1.

(2) Nf(α)*Ng(α) = N(1) = 1 [See Lemma 6, here]

(3) So, Nf(α) = 1 [See Lemma 5, here]

(4) So, Nf(α) = f(α)*f(α²)*...*f(α^λ-1) = 1 [Definition of Norm for Cyclotomic integers, here]

(5) So, f(α)*f(α²)*...*f(α^λ-1) = f(α)*g(α) [Combining step #1 with step #4]

(6) Dividing both sides of step #5 by f(α) gives us the desired result.

QED

Lemma 2: A cyclotomic unit is a factor of any cyclotomic integer.

Proof:

(1) Let f(α) be a unit.

(2) Let h(α) be any cyclotomic integer.

(3) Let g(α) = h(α)*f(α)^-1 [See definition 2 above]

(4) Then, h(α) = g(α)*f(α)

QED

2. Cyclotomic Primes

Definition 3: Irreducible

A cyclotomic integer h(α) is irreducible if for any factorization h(α)=f(α)*g(α), either f(α) or g(α) is a unit.

Definition 4: Cyclotomic Prime

A cyclotomic integer h(α) is prime if:

(a) if h(α) divides f(α)*g(α), then h(α) divides f(α) or g(α).

(b) there exists at least one cyclotomic integer f(α) that h(α) does not divide.

(c) if h(α) is not a factor of f(α) and it is not a factor of g(α), then it is not a factor of f(α)*g(α)

In rational integers, all irreducible nonunits are also primes. One of the question that needs to be addressed is whether this is still the case with cyclotomic integers. I will answer this question in a later blog.

3. More Properties of Units

Lemma 3: a unit * a unit = a unit

Proof:

(1) Let g(α),h(α) be units.

(2) By Lemma 6, here, if i(α)=g(α)*h(α), then Ni(α) = Ng(α)*Nh(α) = 1*1 = 1.

QED

Lemma 4: 1/unit = a unit

Proof:

(1) Let h(α) be a unit

(2) Nh(α) = h(α)*h(α²)*...*h(α^λ-1) = 1. [See Definition 2, here for definition of norm for cyclotomic integers]

(3) From (#2), we can see that:

1/h(α) = h(α²)*...*h(α^λ-1)

(4) Further, we can see that:

Norm(1/h(α)) = Nh(α²)*N(α³)*...*Nh(α^λ-1) = 1*1*1*...*1 = 1 [See Lemma 6, here]

QED

Lemma 5: unit/unit = unit

Proof:

(1) Let h(α),g(α) be units.

(2) h(α)/g(α) = h(α)*(1/g(α))

(3) From Lemma 4 above, we know that (1/g(α)) is a unit.

(4) From Lemma 3 above, we know that a unit*unit = unit.

QED