The contents of today's blog are taken from Harold M. Edwards Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory.

In previous blogs, I have talked about cyclotomic integers and cyclotomic periods derived from cyclotomic integers. In today's blog, I will focus on a single, very important result. That for any prime p where p ≠ λ, there exists an integer f = exponent mod λ (see definition 2, here) and an integer e = (λ - 1)/f (see definition 3, here).

This means that for any prime p, there exist a set of e cyclotomic periods that each have a length of f elements (see here for review of the basic properties of cyclotomic periods).

Lemma 1: x

^{p}- x ≡ (x-1)(x-2)*...*(x-p) ≡ 0 (mod p)

Proof:

(1) Fermat's Little Theorem gives us:

x

^{p-1}≡ 1 (mod p)

So that:

p divides x

^{p-1}- 1

So that:

x(x

^{p-1}- 1) = x

^{p}- x ≡ 0 (mod p)

(2) We also know that:

(x -1)*(x-2)*...(x-p) ≡ 0 (mod p)

(a) There exists a r such that x ≡ r (mod p)

(b) We know that p divides x - r

(c) We also know that r is between 0 and p-1.

(d) if r is 0, then p divides x-p; otherwise, p divides x-r.

(3) Putting (#1) and (#2) together, we have:

x

^{p}- x ≡ (x-1)(x-2)*...*(x-p) ≡ 0 (mod p)

QED

Lemma 2: g(α) = g(α

^{p}) if and only if g(α) is a cyclotomic integer made up of periods of length f.

Proof:

(1) Let λ be an odd prime.

(2) Let α be a primitive root of unity such that α

^{λ}= 1.

(3) Let p be an odd prime distinct from λ

(4) Let τ be mapping such that τα = α

^{p}

(5) Let f be the least positive integer for which p

^{f}≡ 1 (mod λ)

(6) We know that f divides λ -1 (see Lemma 1, here)

(7) Let e= (λ-1)/f

(8) Assume that g(α) is made up of periods of length f. [See here for review of cyclotomic periods]

(9) Then, σ

^{e}g(α) = g(α) [See Lemma 4 here for details.]

(10) Then, there must exist an integer k such that:

τ = σ

^{k}

Since the primitive root (see here for review if needed) can take all possible values mod λ [By definition σ = a mapping between α and α

^{γ}where γ is a primitive root, see here]

(11) Using step #10, we note that:

τ

^{f}= σ

^{kf}

Since τ

^{f}≡ 1 (mod λ), we know that σ

^{kf}≡ 1 (mod λ)

(12) Since the order of λ is λ -1 (see here), we know that kf must be divisible by λ - 1 (see Lemma 2 here) which means it is divisible by ef since ef=λ - 1.

So, ef divides kf means that e must divide k.

(13) So, from step #10 and step #12:

τ is a power of σ

^{e}

(14) So, there exists k' such that ek' = k.

(15) From #14, we have:

τ = (σ

_{1}

^{e})(σ

_{2}

^{e})...(σ

_{k'}

^{e})

and therefore:

τg(α) =(σ

_{1}

^{e})(σ

_{2}

^{e})...(σ

_{k'}

^{e})g(α) = g(α)

(16) Assume that τg(α) = g(α)

(17) We know that there exists k such that:

τ = σ

^{k}[Since σ is a mapping to γ which is a primitive root and the primitive root can take on all values modulo λ]

(18) Since g(α) repeats at τg(α), we know that g(α) consists of e periods of length f. [See Corollary 1.2 here]

(19) By definition, e divides (λ - 1) [See here for definition of periods]

(20) Thus, e is a common divisor of k and λ - 1.

(21) e is the greatest common divisor of k and λ - 1 since:

(a) Let d be an integer that divides both k, λ-1 so that:

k = qd

λ - 1 = df'

(b) Now,

τ

^{f'}= σ

^{kf'}= σ

^{qdf'}= σ

^{(λ - 1)q}which is identity[from step #17, #21a, and since σ

^{λ-1}is the identity, see here]

(c) So that:

f' ≥ f [Since f is least value where p

^{f}≡ 1 (mod λ)]

d ≤ e [Because d = (λ-1)/f' where f' ≥ f and e=(λ - 1)/f]

(22) Using Bezout's Identity, there exists a,b such that:

e = ak + b(λ - 1)

(23) Further,

σ

^{e}= σ

^{ak}σ

^{b(λ-1)}= [from step #22]

= σ

^{ak}= [Since σ

^{b(λ-1)}is identity]

= τ

^{a}[Since τ = σ

^{k}from step #17]

(24) From this, we see that:

σ

^{e}g(α) = τ

^{a}g(α) = g(α)

QED

Theorem 3: For any cyclotomic integer g(α) made up of periods of length f where f is the exponent mod λ for p, there exists an integer u such that g(α) ≡ u (mod p)

Proof:

(1) From Lemma 1 above, we have:

x

^{p}- x ≡ (x-1)(x-2)*...*(x-p) (mod p)

(2) Let x = g(α)

(3) Then:

g(α)

^{p}- g(α) ≡ [g(α) - 1][g(α) - 2]*...*[g(α) - p] (mod p)

(4) From a previous result (see Lemma 3 here), where g(α)

^{p}≡ g(α

^{p}) (mod p), we have:

g(α

^{p}) - g(α) ≡ [g(α) - 1][g(α) - 2]*...*[g(α) - p] (mod p)

(5) Now, since g(α) is made up of periods of length f, we know that g(α

^{p}) = g(α) [See Lemma 2 above]

So that:

g(α

^{p}) - g(α) = 0 ≡ 0 (mod p)

(9) This gives us:

[g(α) - 1][g(α) - 2][g(α) - 3]*...*[g(α)-p] ≡ 0 (mod p)

(10) So at least one of these values must be divisible by p (otherwise, the product of each g(α) - u could not be 0.)

(11) So there exists an integer u such that p divides g(α) - u

Which means that:

g(α) ≡ u (mod p)

QED

Corollary 3.1: For each of the e cyclotomic periods η

_{i}for a given odd prime p, there exists an integer u

_{i}such that η

_{i}≡ u

_{i}(mod p)

Proof:

(1) Each period η

_{i}can be thought of as a cyclotomic integer made up of periods of length f:

g(α) = (0)η

_{0}+ ... + (1)η

_{i}+ ... + (0)η

_{e-1}= η

_{i}

(2) From Theorem 3 above, we know that there exists an integer u

_{i}such that g(α) ≡ u

_{i}(mod p)

(3) So, we see that for each η

_{i}, there exists u

_{i}such that:

η

_{i}≡ u

_{i}(mod p)

QED