Ideal Numbers: Divisors

Not all cyclotomic integers are characterized by unique factorization. In addressing this situation, Ernst Kummer proposed his theory of ideal numbers which attempts to save unique factorization. While unique factorization is saved, there is a price to pay.

The result of Kummer's theory is that it is no longer possible to arbitrarily create a cyclotomic integer based on a set of primes. When using rational primes, you can take any set of primes, multiply them together and get a rational integer. Unfortunately, this is not the case with prime divisors. Not all combinations of prime divisors result in a cyclotomic integer (even though, all cyclotomic integers can be broken down to a unique set of prime divisors).

To better analyze this situation, let me offer some terms taken from Harold M. Edwards' Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory:

Definition 1: Divisor

A divisor is a list of prime divisors including their powers.

For all purposes, you can think about a divisor as a product of prime divisors. A divisor divides a given cyclotomic integer g(α) if and only if the list of prime divisors associated with the divisor divide g(α).

NOTE: Divisor is the same thing as an ideal number. As a convention (following Edwards), I will represent divisors (including prime divisors) as capital letters.

Definition 2: Principal

A divisor D is said to be principal if it can be uniquely associated with a cyclotomic integer g(α) such that g(α) divides another cyclotomic integer h(α) if and only if D divides h(α).

So, to be clear, a combination of prime divisors exist as a cyclotomic integer if and only if the divisor equivalent to the combination of prime divisors is principal. If a divisor is not principal, then the combination of prime divisors which make it up are not uniquely associated with a cyclotomic integer.

To define the class number (link added later), we need one more concept. This one is a bit tricky. Kummer came up with this idea as he was trying to determine under what conditions a divisor is principal.

Kummer's insight is that certain divisors have a special relationship. Let A, B be two such divisors. If AC is principal, then BC is also principal. If AD is not principal, then BD is not principal. In other words, both divisors are principal in exactly the same combinations. Kummer called these two divisors equivalent.

Definition 3: Equivalent Divisors ~

Two divisors A and B are said to be equivalent if for all cases where AC is principal, BC is also principal. That is, A ~ B if and only if AC is principal when and only when BC is principal.

This concept is the basis for the idea of class number which is very important to Kummer's proof. I will talk more about class number in a future blog.

Here are some basic properties of divisors.

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 for a(α)

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

(2) AB is principal → ∃ c(α) such that AB is the divisor for 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 for 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 for 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 (by definition of ~).

(3) Assume AC,BC, AD are principal

(4) Then, AC ~ BC ~ I and AD ~ I [By Lemma 3 above]

(5) (AD)(BC) ~ I [By Lemma 1 and Lemma 3 above]

(6) (AD)(BC) = (BD)(AC) [By Lemma 5 above]

(7) So, since (AC) is principal and (AD)(BC) is principal, it follows that (BD) must also be principal. [By Lemma 2 above]

(8) Thus, we have shown that if there exists a divisor C such that AC is principal and BC is principal, then for any divisor D, if AD is principal, then BD is necessarily also principle.

(9) A~B follows directly from step#(8).

QED