Gauss: Periods of Cyclotomic Equations

In today's blog, I show some major results from Carl Friedrich Gauss in his analysis of periods of cyclotomic equations. These results represent a very small subset of Gauss's work in his classic Disquisitiones Arithmeticae which he wrote when he was 21. These properties of periods of cyclotomic equations are later used to demonstrate Gauss's proof that all cyclotomic polynomials are solvable by radicals.

Today's content is taken straight from Jean-Pierre Tignol's Galois' Theory of Algebraic Equations which covers the history of Galois Theory from a mathematical perspective.

Definition 1: μ_p

Let μ_p denote the set of p-th roots of unity.

Examples:

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

I will use μ_p in the following context:

Definition 2: Q(μ_p)

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

So that:



Definition 3: period of f terms of a p-th root of unity

For any two positive integers: e,f where ef = p-1, the periods of f terms are:

η₀ = ζ₀ + ζ_e + ζ_2e + ... + ζ_e(f-1)

η₁ = ζ₁ + ζ_e+1 + ζ_2e+1 + ... + ζ_e(f-1)+1

η₂ = ζ₂ + ζ_e+2 + ζ_2e+2 + ... + ζ_e(f-1)+2

...

η_e-1 = ζ_e-1 + ζ_2e-1 + ζ_3e-1 + ... + ζ_e(f-1)+(e-1)

If you review Alexander-Theophile Vandermonde's solution of the eleventh root of unity, it is clear that Carl Friedrich Gauss's theory of periods is a generalization of his solution. Interestingly, it is not clear if Gauss derived his solution from the work of Vandermonde or if he came upon it independently as part of his solution of the seventeenth root of unity.

Definition 4: σⁱ(f)

If we number each time that we apply σ such that σ(σ(f)) = σ₁(σ₂(f)), then:
σⁱ(f) = σ₁(σ₂(...(σ_i(f))...))

We will now use the equation ef=p-1 (see Definition 3 above) to define a set whose elements are invariant under σ^e.

Definition 5: K_f

By K_f, let us denote the set of all Q(μ_p) which are invariant under σ^e where ef = p-1.

Examples of K_f:

(1) All rational numbers

u ∈ Q → u ∈ K_f [This is clear from Lemma 6, here]

(2) All periods of cyclotomic equations

This is clear from definition 3 above.

Now, we can use these definitions to identify some properties which we will use later.

Theorem 1: K_f is a vector space

Proof:

The proof follows from Definition 2, here since:

(1) K_f is nonempty [See Lemma 6, here since Q is nonempty]

(2) K_f is closed on addition. [See Lemma 5, here]

(3) K_f is closed on scalar multiplication. [See Lemma 7, here]

(4) K_f addition is associative. [See Lemma 5, here]

(5) 0 ∈ K_f [See Lemma 6, here since 0 ∈ Q]

(6) K_f has negative elements [See Lemma 7, here since (-1)*σ(x) = σ(-x)]

(7) K_f addition is commutative. [See Lemma 5, here]

(8) All elements of K_f are distributive since:

σ(a[b + c]) = σ(ab + ac)

(9) Scalar multiplication is associative [See Lemma 7, here]

(10) Existence of 1 [See Lemma 6, here since 1 ∈ Q]

QED

Theorem 2: Every element in K_f can be written in a unique way as a linear combination with rational coefficients of the e periods of f terms.

(1) Let a be an arbitrary element in K_f [See Definition 5 above]

(2) We can write a as follows: [See Definition 2 above and See Definition 1, here, for ζ_i]

a = a₀ζ₀ + a₁ζ₁ + ... + a_e-1ζ_e-1 +
+ a_eζ_e + a_e+1ζ_e+1 + ... + a_2e-1ζ_2e-1 +
+ ... +
+ a_e(f-1)ζ_e(f-1) + a_e(f-1)+1ζ_e(f-1)+1 + ... + a_p-2ζ_p-2

(3) By the definition of σⁱ [See Definition 4 above], we have:

σ^e(a) = a₀ζ_e + a₁ζ_e+1 + ... + a_e-1ζ_2e-1 +
+ a_eζ2e + a_e+1ζ_2e+1 + ... + a_2e-1ζ_3e-1 +
+ ... +
+ a_e(f-1)ζ₀ + a_e(f-1)+1ζ₁ + ... + a_p-2ζ_e-1.

(4) Since a ∈ K_f, we know that:

σ^e(a) = a

(5) Thus:

a₀ = a_e = a_2e = ... = a_e(f-1)

a₁ = a_e+1 = a_2e+1 = ... = a_e(f-1)+1

...

a_e-1 = a_2e-1 = a_3e-1 = ... = a_p-2

(6) Therefore:

a = a₀(ζ₀ + ζ_e + ... + ζ_e(f-1)) +
+ a₁(ζ₁ + ζ_e+1 + ... + ζ_e(f-1)+1) +
+ ... +
+ a_e-1(ζ_e-1 + ζ_2e-1 + ... + ζ_p-2).

(7) This proves that a is a linear combination of the periods, since the expressions between the brackets are the periods of f terms. [See Definition 3 above]

(8) Further, this expression is unique. [See Theorem 4, here]

QED

Corollary 2.1: 1, η, η₂, ..., η_e-1 is a basis for K_f

Proof:

This follows directly from Theorem 1 above, Theorem 2 above and Lemma 1, here.

QED

Theorem 3:

1, η, η², ..., η^e-1 is a basis for the vector space K_f

Proof:

(1) 1, η, η², ..., η^e-1 are linearly independent [see Definition 1, here for definition of linearly independent if needed] since:

(a) Assume that a₀ + a₁η + ... + a_e-1η^e-1 = 0 for some rational numbers a₀, ..., a_e-1

(b) Then η is the root of the polynomial p(x) where:

p(x) = a₀ + a₁x + ... + a_e-1x^e-1 (from step #1a)

(c) Now if a₀ + a₁η + ... + a_e-1η^e-1 = 0, it follows that:

σ(a₀ + a₁η + ... + a_e-1η^e-1 ) = σ(0) = 0

σ²(a₀ + a₁η + ... + a_e-1η^e-1 ) = σ²(0) = 0

σ³(a₀ + a₁η + ... + a_e-1η^e-1 ) = σ³(0) = 0

...

σ^e-1(a₀ + a₁η + ... + a_e-1η^e-1 ) = σ^e-1(0) = 0

(d) So, σ(η), σ²(η), ..., σ^e-1(η) are all roots of p(x) in step #1b

(e) Now, each of η, σ(η), etc. are the e periods of f terms which are pairwise distinct [See Definition 3 above]

(f) Since by the Fundamental Theorem of Algebra (see Theorem, here), the polynomial p(x) has degree at most e -1, it cannot have as roots the e periods of the f terms unless it is the zero polynomial.

(g) Therefore, a₀ = ... = a_e-1 = 0

(h) This then proves 1, η, η², ..., η^e-1 are linearly independent. [See Definition 1, here]

(2) From Corollary 2.1 above, we know that dim K_f = e. [See Theorem 1, here and Definition 2, here]

(3) But then using the fact 1, η, η², ..., η^e-1 are linearly independent and Lemma 2, here, we can conclude that:

1, η, η², ..., η^e-1 is a basis for K_f.

QED

Corollary 3.1:

If η, η' are periods of f terms, then:

η' = a₀ + a₁η + ... + a_e-1η^e-1

for some rational numbers a₀, ..., a_e-1

Proof:

This follows from Theorem 3 above since η' ∈ K_f and 1, η, η², ..., η^e-1 is a basis for the vector space K_f.

QED

Lemma 4:

if gh=ef=p-1 and f divides g, then it follows that:

K_g ⊂ K_f

Proof:

(1) Since gh=ef and f divides g, there exists an integer k such that:

k = g/f = e/h

(2) Therefore e = hk which gives us that:

σ^e = (σ^h)^k

(3) This means that every element that is invariant under σ^h is also invariant under σ^e since:

(a) Assume that an element a is invariant under σ^h such that:

σ^h(a) = a

(b) Further:

σ^h₁(σ^h₂(...(σ^h_k(a)...))) = a

(4) Since h*k = e, it follows from definition 4 above that:

σ^h₁(σ^h₂(...(σ^h_k(a)...))) = σ^e(a)

(5) And it follows that:

σ^e(a) = a

(6) Since σ^h(a) = a → a ∈ K_g and σ^e(a) = a → a ∈ K_f, it follows that:

K_g ⊂ K_f

QED

Lemma 5:

Let f,g be divisors of p-1.

If f divides g, then every element in K_f is a root of a polynomial of degree g/f with coefficients in K_g

Proof:

(1) Let a be an element of K_f

(2) Let us define k such that:

k = g/f

Since ef = gh, it follows that:

k = g/f = e/h

(3) Let use define P(x) such that:

P(x) = (x - a)(x - σ^h(a))(x - σ^2h(a))*...*(x - σ^h(k-1)(a))

(4) P(x) has degree hk/h = k = g/f

(5) It is also clear that a is a root of P(x). [Since if x=a, then P(x)=0]

(6) We note that:

σ^h(σ^h(k-1)(a)) = σ^hk(a) = (σ^h(a))^k

(7) Since k = e/h, it follows that e=hk and:

(σ^h(a))^k = σ^e(a) = a

(8) Step #3 and step #6 and step #7 give us that:

σ^h(P(x)) = P(x)

(9) Therefore, we conclude that P(x) has coefficients in K_g.

QED

Corollary 5.1:

Let f,g be divisors of p-1 and let η, ξ be periods of f and g terms respectively.

If f divides g, then η is a root of a polynomial of degree g/f whose coefficients are rational expressions of ξ

Proof:

(1) ξ ∈ K_g, and η ∈ K_f

(2) Using Lemma 5 above, we know that η is a root of a polynomial P(x) of degree g/f with coefficients in K_g

(3) Using Theorem 3 above, it follows that:

P(x) has coefficients which are rational expressions of ξ.

QED

References

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

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