Cyclotomic Integers: Principal and Equivalent Divisors

Today's blog is taken straight from Harold M. Edwards' Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory.

When Ernst Kummer proposed his concept "ideal numbers" (what I have been calling divisors as used by Edwards) to "save" the fundamental theorem of arithmetic, he had to give up something. In Kummer's approach, all cyclotomic integers are characterized by a unique divisor which is composed of a unique set of prime divisors. The cost of this approach is that not all compositions of prime divisors are necessarily divisors. For example, in the case of λ = 23, there is no cyclotomic integer which is divisible once by one of the prime divisors of 47 and not divisible by any other prime divisor of 47.

The idea that not all prime divisors and not combinations of prime divisors are divisors of a cyclotomic integer leads the idea of "principal divisors."

Definition 1: principal

A divisor is said to be principal if there is some cyclotomic integer for which it is the divisor.

Example 1: Ordinary integers

In the ordinary integers, all primes are principal in the sense that each prime corresponds to integer.

Example 2: Cyclotomic integers where λ = 23

In this case, none of the prime divisors of 47 are principal since none of them divide a cyclotomic integer exactly once without the cyclotomic integer being divisible by any other prime divisor of 47.

The challenge then becomes to determine under which conditions a divisor is principal. To analyze this issue, Kummer proposed the idea of equivalence which leads the very important idea of class number (which I will describe in a future blog)

Definition 2: Equivalence

Two divisors A and B are said to be equivalent denoted A ~ B if it is true that a divisor of the form AC is principal when and only when BC is principal.

Example 1:

In the case where a divisor is not principal, it requires another divisor in order to principal. So, for example, if AB is principal but A is not, then there exists a set of divisors let us call them B₁, B₂, etc such that AB₁, AB₂, etc. are all principal. In this circumstance, the divisors B₁, B₂, etc. are said to be equivalent.

NOTE:

This concept of equivalence is presented as an idea that helps to explain under what conditions a product of divisors is principal. To help explore these ideas, I will now review some basic properties of principal and equivalent divisors. These again, are taken straight from Edwards.

Lemma 1: If A and B are both principal, then so is AB

Proof:

(1) A is principal → ∃ a(α) such that A is the divisor of a(α)

(2) B is principal → ∃ b(α) such that B is the divisor of b(α)

(3) So, AB is the divisor for a(α)b(α) which means that AB is also principal.

QED

Lemma 2: If A and B are divisors such that A and AB are both principal, then B is principal.

Proof:

(1) A is principal → ∃ a(α) such that A is the divisor of a(α)

(2) AB is principal → ∃ c(α) such that AB is the divisor of c(α)

(3) Since A divides AB, we know that a(α) divides c(α) so that there exists b(α) = c(α)/a(α) and we know that B is the divisor of b(α)

QED

Lemma 3: A is principal if and only if A ~ I where I is the empty divisor, that is the divisor of 1.

Proof:

(1) Assume A is principal

(2) Then there exists a(α) such that A is the divisor of a(α)

(3) In all cases where IC is principal, it is implies that C is principal which means that in all those cases AC is also principal [By Lemma 1 above]

(4) Assume A~I

(5) If IC is principal then C is principal since IC = C.

(6) IC is principal implies that AC principal. [From A ~ I]

(7) So if AC is principal and C is principal then A is principal [By Lemma 2 above]

QED

Lemma 4: The equivalence relation ~ is reflexive, symmetric, and transitive.

Proof:

(1) ~ is reflexive, that is, A ~ A, since A can always be replaced by itself.

(2) ~ is symmetric since multiplication of divisors is commutative. If AB is principal, then BA is also principal.

(3) ~ is transitive since if A~B and B~C, then A~C. Assume there was a divisor D where AD is principal but CD is not, then BD would not be principal (by B~C) and this would be a contradiction of A~B so this divisor D cannot exist.

QED

Lemma 5: Multiplication of divisors is consistent with the equivalence relations. That is A~B implies AC~BC for all divisors C

Proof:

(1) Assume ACD is principal

(2) Then A(CD) is principal since a divisor can be formed from C,D

(3) Then B(CD) is principal (from A~B)

(4) Assume ACD is not principal

(5) Then A(CD) is not principal since a divisor can be formed from C,D

(6) Then B(CD) is not principal (otherwise, we violate A~B)

QED

Lemma 6: Given any divisor A, there is another divisor B such that AB~I

Proof:

(1) N(A) is the divisor of an integer.

(2) Let B = the complement of N(A), that is, N(A)/A.

(3) Then AB is principal and this leads to AB~I (from Lemma 3 above)

QED

Lemma 7: A ~ B implies there exist principal divisors M and N such that AM = BN.

Proof:

(1) Assume A ~ B

(2) There exists C such that AC is principal. (from Lemma 6 above)

(3) AC ~ I (from Lemma 3 above)

(4) Let M = BC

(5) Let N = AC

(6) So AM = ABC = BN

QED

Lemma 8: A ~ B if and only if there is a third divisor C such that AC and BC are both principal.

Proof:

(1) Assume A ~ B

(2) Then there exists a divisor C such that AC is principal (from Lemma 6 above) and BC is principal.

(3) Assume AC,BC are principal

(4) Then A~B by definition.

QED