Cyclotomic Integers: Divisors
In today's blog, I continue to review results 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.
Definition 1: Divisor of a nonzero cyclotomic integer
The divisor of a nonzero cyclotomic integer g(α) is a list, with multiplicities, of all the prime
divisors which divide g(α)
Definition 2: Empty list divisor
A divisor is any finite list of prime divisors. The empty list is considered to be a divisor. The empty list is a the divisor of the cyclotomic integer 1.
Definition 3: product of two divisors
The product of two divisors is the juxtaposition of the two lists counted with multiplicities.
Definition 4: Divisible by a divisor
A cyclotomic integer is said to be divisible by a given divisor if it can be written as a product of that divisor with another divisor. A cyclotomic integer is said to be divisible by a divisor if its divisor is divisible by that divisor and similarly a divisor is said to be divisible by a cyclotomic integer if it is divisible by the divisor of the cyclotomic integer.
NOTE: With these definitions, the fundamental theorem becomes the statement g(α) divides h(α) if and only if the divisor of g(α) divides the divisor of h(α)
Observation: The divisor of a product is the product of the divisors.
Observation: Since 0 is divisible by every nonzero cyclotomic integer, it is sometimes convenient to consider 0 to have the divisor which contains every prime divisor an infinite number of times.
Theorem 1:
Given a prime divisor of p ≠ λ there is a cyclotomic integer ψ(η) made up of periods of length f (the exponent of p mod λ) which is divisible exactly once by that prime divisor of p and which is not divisible by the remaining e-1 prime divisors of p. Consequently, a cyclotomic integer g(α) is divisible with multiplicity μ by the given prime divisor of p if and only if g(α)[σψ(η)]μ[σ2ψ(η)]μ*...*[σe-1ψ(η)]μ is divisible by pμ
Proof:
(1) Let ψ(η) correspond as before to a given prime divisor of p.
That is, let u1, u2, ..., ue be the integers such that ui - ηi is divisible by the given prime divisor and let ψ(η) be the product of ep-e factors j - ηi in which j ≠ ui.
(2) Then ψ(η) is divisible by all e prime divisors of p except the given one.
(3) Let φ(η) = σψ(η) + σ2ψ(η) + ... + σe-1ψ(η).
(4) φ(η) is divisible by the given prime divisor of p (each term of the sum is divisible by it) but not divisible by the remaining e-1 prime divisors of p (for each of these all but one of the terms are divisible by it but that one is not divisible).
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.
(5) If φ(η) is divisible with multiplicity exactly one by the given prime divisor of p, then ψ(η) = φ(η) has the required properties.
(6) If φ(η) is divisible with multiplicty greater than one, then set ψ(η) = φ(η) + p.
(7) Then, ψ(η) is not divisible by the e-1 prime divisors other than the given one (because if it were then φ(η) = ψ(η) - p would be), it is divisible by the given prime divisor of p (because φ(η) is), and it is not divisible with multiplicity greater than one (because if it were then p = ψ(η) - φ(η) would be).
(8) This completes the construction of ψ(η).
(9) The second statement of the theorem is an immediate consequence of the fundamental theorem.
QED
Theorem 2:
The cyclotomic integers called "prime" in the examples earlier are indeed prime. More generally, if p ≠ λ is a prime whose exponent mod λ is f and if g(α) is a cyclotomic integer whose norm is pf then g(α) is prime. [If g(α) = g(η) is made up of periods of length f then the condition Ng(α) = pf can also be stated in the form g(η)*σg(η)*...*σe-1g(η) = ± p.]
Proof:
(1) Let ν1, ν2, ...., νe be the exact multiplicities with which the prime divisors of p divide g(α).
(2) Then σg(α) is divisible with the multiplicities ν2, ν3, ...., νe, ν1 exactly.
NOTE: This is true since σg(α) just shifts the prime divisors. (See Theorem 4, here)
(3) Then σ2g(α) is divisible with the multiplicities ν3, ν4, ...., ν1, ν2 exactly and so forth.
NOTE: Same reason as in step #2.
(4) Therefore, each prime divisor of p divides Ng(α) with multiplicity exactly (ν1 + ν2 + ... + νe)f.
NOTE: We know this since in the given, Ng(α) = pf.
(5) If Ng(α) = pf, then one νi must be 1 and the others 0.
NOTE: We know this since by our given (ν1 + ν2 + ... + νe)f= f since Ng(α) = pf.
(6) Then, by the fundamental theorem, divisibility by g(α) coincides with divisibility by a prime divisor and it follows that g(α) is prime.
QED
Definition 5: greatest common divisor of p
Let a prime divisor of p be given and let ψ(η) be a cyclotomic integer as in the theorem above such that it is made up of periods of length f (the exponent of p mod λ) which is divisible just once by the prime divisor of p in question and not divisible by any of the remaining e-1 prime divisors of p. Then the given prime divisor wil be designated (p,ψ(η)). Since this divisor divides p and ψ(η), both with multiplicity exactly one, and no other prime divisor divides both, (p,ψ(η)) is the greatest common divisor of p and ψ(η).
Definition 6: Empty Divisor
The empty divisor which divides 1 will be denoted by I and an exponent of 0 on a divisor A will be defined to mean A0 = I.
Definition 1: Divisor of a nonzero cyclotomic integer
The divisor of a nonzero cyclotomic integer g(α) is a list, with multiplicities, of all the prime
divisors which divide g(α)
Definition 2: Empty list divisor
A divisor is any finite list of prime divisors. The empty list is considered to be a divisor. The empty list is a the divisor of the cyclotomic integer 1.
Definition 3: product of two divisors
The product of two divisors is the juxtaposition of the two lists counted with multiplicities.
Definition 4: Divisible by a divisor
A cyclotomic integer is said to be divisible by a given divisor if it can be written as a product of that divisor with another divisor. A cyclotomic integer is said to be divisible by a divisor if its divisor is divisible by that divisor and similarly a divisor is said to be divisible by a cyclotomic integer if it is divisible by the divisor of the cyclotomic integer.
NOTE: With these definitions, the fundamental theorem becomes the statement g(α) divides h(α) if and only if the divisor of g(α) divides the divisor of h(α)
Observation: The divisor of a product is the product of the divisors.
Observation: Since 0 is divisible by every nonzero cyclotomic integer, it is sometimes convenient to consider 0 to have the divisor which contains every prime divisor an infinite number of times.
Theorem 1:
Given a prime divisor of p ≠ λ there is a cyclotomic integer ψ(η) made up of periods of length f (the exponent of p mod λ) which is divisible exactly once by that prime divisor of p and which is not divisible by the remaining e-1 prime divisors of p. Consequently, a cyclotomic integer g(α) is divisible with multiplicity μ by the given prime divisor of p if and only if g(α)[σψ(η)]μ[σ2ψ(η)]μ*...*[σe-1ψ(η)]μ is divisible by pμ
Proof:
(1) Let ψ(η) correspond as before to a given prime divisor of p.
That is, let u1, u2, ..., ue be the integers such that ui - ηi is divisible by the given prime divisor and let ψ(η) be the product of ep-e factors j - ηi in which j ≠ ui.
(2) Then ψ(η) is divisible by all e prime divisors of p except the given one.
(3) Let φ(η) = σψ(η) + σ2ψ(η) + ... + σe-1ψ(η).
(4) φ(η) is divisible by the given prime divisor of p (each term of the sum is divisible by it) but not divisible by the remaining e-1 prime divisors of p (for each of these all but one of the terms are divisible by it but that one is not divisible).
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.
(5) If φ(η) is divisible with multiplicity exactly one by the given prime divisor of p, then ψ(η) = φ(η) has the required properties.
(6) If φ(η) is divisible with multiplicty greater than one, then set ψ(η) = φ(η) + p.
(7) Then, ψ(η) is not divisible by the e-1 prime divisors other than the given one (because if it were then φ(η) = ψ(η) - p would be), it is divisible by the given prime divisor of p (because φ(η) is), and it is not divisible with multiplicity greater than one (because if it were then p = ψ(η) - φ(η) would be).
(8) This completes the construction of ψ(η).
(9) The second statement of the theorem is an immediate consequence of the fundamental theorem.
QED
Theorem 2:
The cyclotomic integers called "prime" in the examples earlier are indeed prime. More generally, if p ≠ λ is a prime whose exponent mod λ is f and if g(α) is a cyclotomic integer whose norm is pf then g(α) is prime. [If g(α) = g(η) is made up of periods of length f then the condition Ng(α) = pf can also be stated in the form g(η)*σg(η)*...*σe-1g(η) = ± p.]
Proof:
(1) Let ν1, ν2, ...., νe be the exact multiplicities with which the prime divisors of p divide g(α).
(2) Then σg(α) is divisible with the multiplicities ν2, ν3, ...., νe, ν1 exactly.
NOTE: This is true since σg(α) just shifts the prime divisors. (See Theorem 4, here)
(3) Then σ2g(α) is divisible with the multiplicities ν3, ν4, ...., ν1, ν2 exactly and so forth.
NOTE: Same reason as in step #2.
(4) Therefore, each prime divisor of p divides Ng(α) with multiplicity exactly (ν1 + ν2 + ... + νe)f.
NOTE: We know this since in the given, Ng(α) = pf.
(5) If Ng(α) = pf, then one νi must be 1 and the others 0.
NOTE: We know this since by our given (ν1 + ν2 + ... + νe)f= f since Ng(α) = pf.
(6) Then, by the fundamental theorem, divisibility by g(α) coincides with divisibility by a prime divisor and it follows that g(α) is prime.
QED
Definition 5: greatest common divisor of p
Let a prime divisor of p be given and let ψ(η) be a cyclotomic integer as in the theorem above such that it is made up of periods of length f (the exponent of p mod λ) which is divisible just once by the prime divisor of p in question and not divisible by any of the remaining e-1 prime divisors of p. Then the given prime divisor wil be designated (p,ψ(η)). Since this divisor divides p and ψ(η), both with multiplicity exactly one, and no other prime divisor divides both, (p,ψ(η)) is the greatest common divisor of p and ψ(η).
Definition 6: Empty Divisor
The empty divisor which divides 1 will be denoted by I and an exponent of 0 on a divisor A will be defined to mean A0 = I.
posted by Larry Freeman at 1:58 AM
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