In a previous blog, I spoke a good deal about ideal numbers but I avoided defining them. Instead, I've defined congruence modulo an ideal number and divisible by an ideal number. But, what exactly is an ideal number in and of itself? This is what I will define in today's blog.
Most of the content in today's blog is taken from Harold M. Edwards Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory.
Edwards argues that "ideal number" is a misleading term because it begs the question "what is an ideal number." He prefers the term "divisor." Throughout my blog, I will use both terms.
In order to define ideal numbers, I will need to use the following thereom:
Theorem 1: There exists a cyclotomic integer φ(η) such that φ(η) is divisible exactly once by a given prime divisor P that divides a prime p but is not divisible by any other prime divisor of p.
Proof:
(1) We can assume that p ≠ λ since it p = λ this theorem is clearly met with α - 1 as the cyclotomic integer.
(2) Let ψ(η) be the formula constructed as before from η1, ..., ηe and u1, ..., ue where e = (λ - 1)/f where f is the exponent mod λ for p. [See here for definition of exponent mod λ]
(3) As was shown before, ψ(η) is divisible by all but one of the e prime divisors of p.
(4) Let φ(η) = σψ(η) + σ2ψ(η) + ... + σe-1ψ(η).
(5) φ(η) is now divisible only by one prime divisor of p (the one that didn't divide ψ(η) and is not divisible by any of the other prime divisors.
We know this is true in since in each case, all but one of the values is divisible by a prime divisor of p. So, if it divides all but the first element that result is:
nonzero + 0 + 0 + ... + 0 ≡ nonzero (mod a prime divisor of p)
On the other hand for the prime divisor that doesn't divide ψ(η), we know that all elements of the summation are divisible by this prime divisor.
NOTE: The reason this is true is that the σ function shifts the ηi value. So while uj - ηi is included in ψ(η), when we apply σ to it, we get uj - ηj. In other words, since we are including all possible shifts as part of the sum, one of these shifts results in a difference which is divisible by p.
(6) If φ(η) is divisible exactly once by the given prime divisor of p, then φ(η) has the required properties and we are done.
(7) If φ(η) is divisible more than once, then φ(η) + p has the property that we are looking for.
(a) Only one of the prime divisors of p divides φ(η) + p
Since each of the prime divisors of p divides p (see Theorem, here), we know that all but one cannot divide φ(η) + p [If any other prime divisors divides φ(η) + p, then it would also divide φ(η) since it divides p. But this is not the case as shown in step #4]
(b) It is only divisible once by the prime divisor of p in question
This is true because the prime divisor of p only divides p once [Since there are e distinct prime divisors that divide up p, see Lemma 1, here]
If the prime divisor of p divides φ(η) + p more than once, then it would divide p more than once (since it divides φ(η) more than once) which it does not.
QED
Definition 1: Prime Divisor P of p
A prime divisor P that divides a rational prime p is defined as gcd(p,φ(η)) where φ(η) is the cyclotomic integer that satisfies the theorem above.
Observation:
A prime divisor is the greatest common denominator between p and the cyclotomic integer φ(η). It may or may not correspond to a real cyclotomic prime. When it corresponds to a cyclotomic prime, we say that the prime divisor is principal (see here for more on principal ideal numbers).
In my next blog, I will talk about the norm of ideal numbers. To do this, I will need to define a version of the σ permutation for ideal numbers.
Definition 2: σP
σP = σ(p,ψ(η)) = (p,σψ(η))
Observation:
We can see that the σ transformation changes a given prime divisor P1 of p into another prime divisor, say, P2 of p.
Definition 3: ideal number
An ideal number is the product of powers of prime divisors .
Observation:
We can see that just like a prime divisor, an ideal number may correspond to a real cyclotomic integer or it may not. When it corresponds to a real cyclotomic integer, we say that the ideal number is principal.
Definition 4: σ for ideal numbers
For an ideal number A:
σA = σ(P1a*P2b*...*Pnc) = σ(P1a)*σ(P2b)*...*σ(Pnc) =
= (σP1)a*(σP2)b*...*(σPnc)
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