Gauss: σ notation

In today's blog, I present a mapping notation σ(f) that I will use in proofs about periods of cyclotomic equations. I will talk in more detail in my next blog about Gauss's concept of periods which generalize the same method that Alexander-Theophile Vandermonde used to solve the eleventh root of unity.

The content in today's blog is taken straight from Jean-Pierre Tignol's Galois' Theory of Algebraic Equations.

Lemma 1:

for any prime p, if m ≡ n (mod p), and ζ is a p-th root of unity

then:

ζ^m = ζⁿ

Proof:

(1) Assume m ≡ n (mod p) [See here for a review of modular arithmetic if needed]

(2) Then, there exists an integer d such that:

0 ≤ d ≤ p-1

and

m ≡ d (mod p)
n ≡ d (mod p)

(3) So there exists m' and n' such that:

m = m'*p + d
n = n'*p + d

(4) Since ζ^p = 1 (see here for review of roots of unity if needed), this gives us that:

ζ^m = ζ^{m'*p + d} = (ζ^p)^m'*ζ^d = 1^m'*ζ^d = ζ^d

ζⁿ = ζ^{n'*p + d} = (ζ^p)^n'*ζ^d = 1^n'*ζ^d = ζ^d

QED

Definition 1: ζ_i

Let ζ_i = ζ^gi where g is a primitive root of a prime p.

Lemma 2:

ζ_p-1 = ζ₀

ζ_p = ζ₁

Proof:

(1) Since g is a primitive root, g^p-1 ≡ 1 (mod p) [By Fermat's Little Theorem, see here].

(2) So using Lemma 1 above, it follows that:

ζ_p-1 = ζ^gp-1 = ζ¹ = ζ^g0 = ζ₀

and

ζ_p = ζ^g(p-1)+1 = ζ^gp-1*g1 = (ζ^gp-1)^g = (ζ¹)^g = ζ^g1 = ζ₁

QED

Definition 2: μ_p

Let μ_p denote the set of p-th roots of unity so that:

μ_p = { 1, ζ₀, ζ₁, ..., ζ_p-2 }

Example:

μ₁ = {1}
μ₂ = {1, -1}
μ₃ = {1, (1/2)[-1 + √-3], 1/2)[-1 - √-3])
μ₄ = {1, -1, i, -i}

Definition 3: σ(ζ) where ζ ∈ μ_p

Let σ be a map that changes f(ζ) to f(ζ^g)

Lemma 3: σ(ζ_i) = ζ_i+1

Proof:

(1) From the definition of ζ_i [See Definition 1 above]

σ(ζ_i) = σ(ζ^gi)

(2) From the definition of σ [See Definition 2 above]

σ(ζ^gi) = ζ^gi+1 = ζ_i+1

QED

Lemma 4:

if ρ, ω ∈ μ_p

then:

σ(ρω) = σ(ρ)σ(ω)

Proof:

(1) Since ρ, ω ∈ μ_p, there exists i,j such that:

ρ = ζ_i

ω = ζ_j

(2) σ(ρ)σ(ω) = (ζ^gi+1)*(ζ^gj+1) = (ζ^gi)^g*(ζ^gj)^g = (ζ^gi*ζ^gj)^g

(3) There also exists a,b such that:

gⁱ ≡ a (mod p)
g^j ≡ b (mod p)

(4) So that ρ*ω = ζ^a*ζ^b = ζ^a+b

(5) There exists d such that a+b ≡ d (mod p) and 0 ≤ d ≤ p-1 so it follows that ζ^d ∈ μ_p

(6) There exists k such that g^k ≡ d (mod p) so we have:

σ(ρ*ω) = σ(ζ^gk) = ζ^gk+1 = (ζ^gk)^g = (ζ^{a + b})^g = (ζ^{gi + gj})^g = (ζ^gi*ζ^gj)^g

QED

Definition 4: Q(μ_p)

Let Q(μ_p) denote the set of complex numbers that are rational expressions in these p-th roots of unity.

This gives us that:



Definition 5: σ(f) where f ∈ Q(μ_p)

σ(a₀ζ₀ + ... + a_p-2ζ_p-2) = a₀σ(ζ₀) + ... + a_p-2σ(ζ_p-2)

where a_i ∈ Q and ζ_i ∈ μ_p

Lemma 5: σ is well-defined on the whole of Q(μ_p)

Proof:

This follows from Definition 5 above and Theorem 4, here.

QED

Lemma 6:

The map σ is a field automorphism of Q(μ_p) which leaves every element of Q invariant.

Proof:

(1) σ is bijective . [See Definition 1, here for definition of bijective; see Definition 5 above]

(2) σ(ua + vb) = uσ(a) + vσ(b) for a,b ∈ Q(μ_p) and u,v ∈ Q. [see Definition 5 above]

(3) If a ∈ Q, then using Corollary 1.1, here, we have:

a = (-a)ζ + (-a)ζ² + ... + (-a)ζ^p-1

where ζ is a primitive p-th root of unity

(4) Since each of these ζⁱ corresponds to a different p-th root of unity (see Theorem 3, here), this implies that:

a = (-a)ζ₀ + (-a)ζ₁ + ... + (-a)ζ_p-2

(5) This shows that every rational number is invariant under σ.

(6) Finally: σ(ab) = σ(a)σ(b) where a,b ∈ Q(μ_p) since:

(a) We can define a,b, and ab as summations:

a = ∑ (i=0, p-2) a_iζ_i

b = ∑ (j=0, p-2) b_jζ_j

ab = ∑ (i,j =0, p-2) a_ib_jζ_iζ_j

(b) From Definition 5 above, we have:

σ(ab) = ∑ (i,j=0, p-2) σ(a_ib_jζ_iζ_j)

(c) Since a_i, b_j ∈ Q, we have:

σ(ab) = ∑ (i,j=0, p-2) a_ib_jσ(ζ_iζ_j)

(d) Since ζ_i, ζ_j ∈ μ_p, using Lemma 4 above, we have:

σ(ab) = ∑ (i,j=0, p-2) a_ib_jσ(ζ_i)σ(ζ_j)

(e) Also, using Definition 5 above, we have:

σ(a)σ(b) = [∑ (i=0, p-2) σ(a_iζ_i)][∑ (j=0, p-2) σ(b_jζ_j)]

(f) Since a_i, b_j ∈ Q, we have:

σ(a)σ(b) = [∑ (i=0, p-2) a_iσ(ζ_i)][∑ (j=0, p-2) b_jσ(ζ_j)] =

= ∑ (i,j=0, p-2) a_ib_jσ(ζ_i)σ(ζ_j)

(7) This shows that σ is a field automorphism of Q(μ_p) [See Definition 6, here, for definition of field automorphism]

QED

Definition 6: σ(f) where f ∈ Q(μ_k)(μ_p) where k divides p-1.

σ(a₀ζ₀ + ... + a_p-2ζ_p-2) = a₀σ(ζ₀) + ... + a_p-2σ(ζ_p-2)

where a_i ∈ Q(μ_k) and ζ_i ∈ μ_p

Lemma 7: σ is well-defined on the whole of Q(μ_k)(μ_p)

Proof:

This follows from Definition 6 above and Corollary 3.1, here.

QED

Lemma 8:

The map σ is a field automorphism of Q(μ_k)(μ_p) which leaves every element of Q(μ_k) invariant.

Proof:

(1) σ is bijective . [See Definition 1, here for definition of bijective; see Definition 6 above]

(2) σ(ua + vb) = uσ(a) + vσ(b) for a,b ∈ Q(μ_k)(μ_p) and u,v ∈ Q(μ_k). [see Definition 6 above]

(3) If a ∈ Q(μ_k), then using Corollary 1.1, here, we have:

a = (-a)ζ + (-a)ζ² + ... + (-a)ζ^p-1

where ζ is a primitive p-th root of unity

(4) Since each of these ζⁱ corresponds to a different p-th root of unity (see Theorem 3, here), this implies that:

a = (-a)ζ₀ + (-a)ζ₁ + ... + (-a)ζ_p-2

(5) This shows that every number a ∈ Q(μ_k) is invariant under σ.

(6) Finally: σ(ab) = σ(a)σ(b) where a,b ∈ Q(μ_k)(μ_p) since:

(a) We can define a,b, and ab as summations:

a = ∑ (i=0, p-2) a_iζ_i

b = ∑ (j=0, p-2) b_jζ_j

ab = ∑ (i,j =0, p-2) a_ib_jζ_iζ_j

(b) From Definition 6 above, we have:

σ(ab) = ∑ (i,j=0, p-2) σ(a_ib_jζ_iζ_j)

(c) Since a_i, b_j ∈ Q, we have:

σ(ab) = ∑ (i,j=0, p-2) a_ib_jσ(ζ_iζ_j)

(d) Since ζ_i, ζ_j ∈ μ_p, using Lemma 4 above, we have:

σ(ab) = ∑ (i,j=0, p-2) a_ib_jσ(ζ_i)σ(ζ_j)

(e) Also, using Definition 6 above, we have:

σ(a)σ(b) = [∑ (i=0, p-2) σ(a_iζ_i)][∑ (j=0, p-2) σ(b_jζ_j)]

(f) Since a_i, b_j ∈ Q(μ_k), we have:

σ(a)σ(b) = [∑ (i=0, p-2) a_iσ(ζ_i)][∑ (j=0, p-2) b_jσ(ζ_j)] =

= ∑ (i,j=0, p-2) a_ib_jσ(ζ_i)σ(ζ_j)

(7) This shows that σ is a field automorphism of Q(μ_k)(μ_p) [See Definition 6, here, for definition of field automorphism]

QED

References

• Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, 2001