## Wednesday, August 02, 2006

### Ideal Numbers: Cyclotomic primes that divide a prime p ≠ λ

In a previous blog, I spoke about the properties of α - 1 and I showed that α - 1 is the only cyclotomic prime that divides λ. In today's blog, I will talk about properties of cyclotomic primes that divide a standard prime p such that p ≠ λ.

The content of today's blog is based on Harold M. Edwards' Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory.

Lemma 1: For every cyclotomic prime h(α), there exists a rational prime p such that h(α) divides p.

Proof:

(1) Let h(α) be a cyclotomic prime. [See here for a review of cyclotomic primes]

(2) We know that Nh(α) is a rational integer. [See Lemma 5, here]

(3) By the Fundamental Theorem of Arithmetic, Nh(α) consists of a set of rational primes: p1*...*pn.

(4) Since h(α) is a cyclotomic prime, it must divide one of these primes pi. [See Definition 4, here]

QED

Lemma 2: Every cyclotomic integer g(α) is congruent mod h(α) to a cyclotomic integer of the form a1αf-1 + a2αf-2 + ... + af where ai are positive integers less than a prime integer p and h(α) is a prime that divides p.

Proof:

(1) From a previous result, we know that every cyclotomic integer g(α) can be expressed in the following form:

g(α) = g1(η)αf-1 + g2(η)αf-2 + ... + gf(η)

where gi(η) are cyclotomic integers made up of periods of length f. [See Corollary 3.1, here]

(2) Each cyclotomic integer gi(η) is congruent mod h(α) to an integer gi(u) where gi(u) denotes the integer obtained by substituting u1 for η1, u2 for η2, etc. [See Corollary 2.1, here]

(3) We know that there exists a rational prime p such that h(α) divides p. [See Lemma 1 above]

(4) We can assume that gi(u) is between 0 and p-1 since:

(a) Assume that gi(u) ≥ p.

(b) There exists a value i such that gi(u) ≡ i (mod p) and i is between 0 and p-1.

(c) But then gi(u) ≡ i (mod h(α)) since h(α) divides p.

(d) So, we can conclude that gi(η) ≡ i (mod h(α))

(5) So that putting it all together, we have:

g(α) = g1(η)αf-1 + g2(η)αf-2 + ... + gf(η) ≡ g1(u)αf-1 + g2(u)αf-2 + ... + gf(u) ≡

≡ a
1αf-1 + a2αf-2 + ... + af (mod h(α))

where ai is between 0 and p-1.

QED

Corollary 2.1: Every cyclotomic integer g(α) is congruent mod h(α) to 1 of pf specific cyclotomic integers.

Proof:

(1) From Lemma 2 above, every integer g(α) satisfies the equation below:

g(α) ≡ a1αf-1 + a2αf-2 + ... + af (mod h(α)) where ai is between 0 and p-1.

(2) Now the conclusion follows from noting that each of the f different integers can take p different values which gives us: p*p*...*p = pf different values.

QED

An additive group is a group defined around the operation of addition. [See here for a review of the concept of a group]

Example: Z9 is an additive group

Z9 = { 0, 1, 2, 3, 4, 5, 6, 7, 8 }

It is clear that it has all the properties of a group:

(1) Closure: addition of any two integers modulo 9 results in another integer modulo 9.

(2) Identity: 0 is the identity.

(3) Inverse: For any integer, 9-i is the inverse. For example, 1 + 8 = 0 modulo 9.

(4) Associativity: For any a,b,c ∈ Z9, we see that:

a + (b + c) = (a + b) + c

Lemma 3: The additive group of cyclotomic integers mod p has pλ-1 elements.

Proof:

(1) Let λ be an odd prime and let α be a root of unity such that αλ = 1 but for all positive integers i less than λ, αi ≠ 1. [See here for review of roots of unity]

(2) All cyclotomic integers based on λ can be put into this form:

a0 + a1α + a2α2 + ... + aλ-1αλ-1

where ai are all integers [See Lemma 1 here]

(3) Since we are talking about values modulo p, we can assume that ai is between 0 and p-1.

(4) This means that there are λ-1 elements that can take values of 0 to p-1.

(5) If we count all possible values, this leads us to λ - 1 multiples:

[0..p-1]*[0..p-1]*...*[0..p-1] = pλ-1

QED

Lemma 4: if h(α) is a cyclotomic prime that divides a rational prime p, then the additive group of cyclotomic integers mod h(α) is a subgroup of the additive group of cyclotomic integers mod p.

Proof:

(1) We know that the set of cyclotomic integers mod p under '+' is an abelian group since:

(a) It is closed on the operation of '+'

(b) 0 mod p is the identity element.

(c) For any cyclotomic integer ≡ r (mod p), the inverse element is p-r.

(d) '+' is clearly associative in nature.

(e) It is abelian since '+' is commutative.

(2) We can make the same arguments to show that the cyclotomic integers mod h(α) is an abelian group on the operation of addition.

(3) To complete this proof, we only need to show that the set of cyclotomic integers mod h(α) is a subset of the cyclotomic integers mod p.

(4) This is the case since:

(a) Let g(α) be a cyclotomic integer.

(b) Then, there exists r(α) such that g(α) ≡ r(α) (mod h(α)) so that r(α) ∈ additive group of cyclotomic integers mod h(α)

(c) Now we can assume r(α) has the following form (see Lemma 1, here):

a0 + a1α + ... + aλ-1αλ-1

(d) We can further assume that all ai are between 0 and p-1 since if ai is greater than p, then there exists a' such that ai ≡ a' (mod p) where a' is less than p and further if ai ≡ a' (mod p), then ai ≡ a' (mod h(α)) because h(α) divides p.

(e) But if all ai are between 0 and p-1, then r(α) ∈ the additive group of cyclotomic integers mod p.

QED

Definition 2: Exponent of p mod λ

The exponent of p mod λ is the smallest positive integer whereby pf ≡ 1 (mod λ)

Lemma 5: For each cyclotomic prime h(α) that divides a standard prime p ≠ λ, there exists a set of pf incongruent classes mod h(α) where f is the exponent mod λ

Proof:

(1) The number of incongruent elements mod h(α) is a power of p, say pn since:

(a) The additive group of cyclotomic integers mod h(α) is a subgroup of the additive group of cyclotomic integers mod p. [See Lemma 4 above]

(b) The additive group of cyclotomic integers mod p has pλ-1 elements. [See Lemma 3 above]

(c) Since the additive group of cyclotomic integers mod h(α) is a subgroup of the additive group of cyclotomic integers mod p, the order of the first group must divide pλ - 1. [By Lagrange's Theorem, see here]

(d) Because p is prime, the number of elements in the additive group of cyclotomic integers mod h(α) must be a power of p.

(2) The number of incongruent cyclotomic integers mod h(α) is at least λ + 1 because 0, α, α2, ..., αλ=1 are all incongruent mod h(α) since:

(a) Any cyclotomic integer divisible by h(α) must have a norm divisible by p [since p = Nh(α), see Lemma 6 here, and since h(α) divides g(α) → Nh(α) divides Ng(α), see Lemma 6 here]

(b) On the other hand, monomials αj - 0 have norm = 1 [See Lemma 5, here]

(c) The binomials αi - αj (where i is not congruent to j mod λ) have a norm equal to N(α-1)=λ [See Lemma 6, here]

(d) Neither 1 nor λ is divisible by p, so none of these cyclotomic integers are divisible by h(α)

(3) The number of nonzero incongruent cyclotomic integers mod h(α) is divisible by λ since:

(a) If α, α2, ..., αλ =1 are all the nonzero cyclotomic integers mod h(α), then there are exactly λ of them.

(b) Assume that there exists a cyclotomic integer ψ(α) such that ψ(α) is not congruent to 0 mod h(α) and ψ(α) is not congruent to αj mod h(α) for j = 1,2, ..., λ.

(c) Then ψ(α)α, ψ(α)α2, ..., ψ(α)αλ = ψ(α) are all nonzero mod h(α) [because h(α) is prime] and distinct mod h(α) [because ψ(α)αj ≡ ψ(α)αk would imply αj ≡ αk) and distinct from α, α2, ..., αλ (because ψ(α)αj ≡ αi would imply ψ(α) ≡ αk)

(d) If this is all the all the possible nonzero cyclotomic integers congruent to h(α), then there are of them.

(e) Eventually, we run out of possibilities. Let us say that this happens after m such iterations of step (#3c).

(f) Then, there are distinct nonzero cyclotomic integers mod h(α)

(4) From step #1, we get that the total number of nonzero distinct cyclotomic integers mod h(α) is pn - 1.

(5) So putting #4 with #3, we get:

mλ = pn - 1.

(6) This gives us that:

pn ≡ 1 ( mod λ)

(7) Since f is the exponent of p mod λ, (see definition 2 above), we know that n ≥ f. [We know that f ≠ 0 since pn is greater than λ + 1. ]

(8) Thus, the number of pn of incongruent elements mod h(α) is at least pf.

QED