Gauss: Proof that all roots of unity are solvable by radicals

After Carl Friedrich Gauss had solved the seventeenth root of unity, he proceed to generalize this result to show that all roots of unity are solvable by radicals. In today's blog, I will give Gauss's proof which was first presented as part of Gauss's classic work Disquisitiones Arithmeticae when Gauss was 21.

Today's content is taken straight from Jean-Pierre Tignol's Galois' Theory of Algebraic Equations.

In the proofs below, I use σ(f) where f ∈ Q(μ_k)(μ_p) which is defined in Definition 6, here. For definition of K_f, see Definition 5, here.

Lemma 1:

Let ζ be a p-th period of unity.

Let ef=p-1

Let η_i = ζ_i + ζ_e+i + ζ_2e+i + ... + ζ_e(f-1)+i

Then:

σ^e(η_i) = η_i

Proof:

σ^e(η_i) = σ^e(ζ_i + ζ_e+i + ζ_2e+i + ... + ζ_e(f-1)+i) =
=ζ_e+i + ζ_2e+i + ζ_3e+i + ... + ζ_ef+i =
= ζ_e+i + ζ_2e+i + ... + ζ_p-1+i =
= ζ_e+i + ζ_2e+i + ... + ζ_e(f-1)+i + ζ_0+i = η_i

QED

Corollary 1.1:

Let ζ be a p-th period of unity.

Let ef=p-1

Let η_i = ζ_i + ζ_e+i + ζ_2e+i + ... + ζ_e(f-1)+i

Let a ∈ Q

Then:

σ^e(aη_i) = aη_i

Proof:

σ^e(aη_i) = σ^e(a[ζ_i + ζ_e+i + ζ_2e+i + ... + ζ_e(f-1)+i ]) =

= σ^e(aζ_i + aζ_e+i + aζ_2e+i + ... + aζ_e(f-1)+i ) =

= aζ_e+i + aζ_2e+i + aζ_3e+i + ... + aζ_ef+i =

= aζ_e+i + aζ_2e+i + aζ_3e+i + ... + aζ_p-1+i =

= aζ_e+i + aζ_2e+i + ... + aζ_e(f-1)+i + aζ_0+i = aη_i

QED

Lemma 2:

Let ζ be a p-th period of unity.

Let ef=p-1

Let η_i = ζ_i + ζ_e+i + ζ_2e+i + ... + ζ_e(f-1)+i

Then if a_i ∈ Q:

σ^e(a₀η₀ + a₁η₁ + a₂η₂ + ... + a_e-1η_e-1) = a₀η₀ + a₁η₁ + a₂η₂ + ... + a_e-1η_e-1

Proof:

(1) σ^e(a₀η₀ + a₁η₁ + a₂η₂ + ... + a_e-1η_e-1) = σ^e(a₀η₀) + σ^e(a₁η₁) + σ^e(a₂η₂) + ... + σ^e(a_e-1η_e-1) [From Definition 5, here]

(2) From Corollary 1.1 above, we have:

σ^e(a₀η₀) + σ^e(a₁η₁) + σ^e(a₂η₂) + ... + σ^e(a_e-1η_e-1) = a₀η₀ + a₁η₁ + a₂η₂ + ... + a_e-1ηe-1

QED

Corollary 2.1:

if f(ζ), g(ζ) ∈ K_f, then: f(ζ)*g(ζ) ∈ K_f

Proof:

(1) Using Theorem 2, here, there exists a₀, ..., a_e-1 and b₀, ..., b_e-1 such that:

f(ζ) = a₀η₀ + a₁η₁ + a₂η₂ + ... + a_e-1η_e-1

g(ζ) = b₀η₀ + b₁η₁ + b₂η₂ + ... + b_e-1η_e-1

(2) σ^e(f(ζ)g(ζ)) = σ^e(f(ζ))σ^e(g(ζ)) [See Lemma 6, here]

(3) σ^e(f(ζ))σ^e(g(ζ)) = σ^e(a₀η₀ + a₁η₁ + a₂η₂ + ... + a_e-1η_e-1)σ^e(b₀η₀ + b₁η₁ + b₂η₂ + ... + b_e-1η_e-1) = (a₀η₀ + a₁η₁ + a₂η₂ + ... + a_e-1η_e-1)(b₀η₀ + b₁η₁ + b₂η₂ + ... + b_e-1η_e-1) =

= f(ζ)g(ζ) [This follows directly from Lemma 2 above]

QED

Lemma 3

Let ζ be a p-th period of unity.

Let ef=gh=p-1 where f divides g and k=g/f

Let η_i = ζ_i + ζ_e+i + ζ_2e+i + ... + ζ_e(f-1)+i

Let ω be a k-th root of unity

Then:

(η₀ + ωη_h + ... + ω^k-1η_h(k-1))^k = a₀η₀ + ... + a_e-1η_e-1

where the coefficients a₀, ..., a_e-1 are rational expressions in ω over Q [that is, they are rational expressions in Q(μ_k)]

Proof:

(1) I will show that (η₀ + ωη_h + ... + ω^k-1η_h(k-1))^k can be expressed as

a₀η₀ + ... + a_e-1η_e-1 where the coefficients a₀, ..., a_e-1 are rational expressions in ω over Q

(2) Assume that k = 1, then it is clear that a₀ = 1, a₁ = ω, a₂ = ω², ..., a_k-1 = ω^k-1

(3) Assume that our hypothesis in step #2 is true up to n such that k ≤ n, then the hypothesis holds for (η₀ + ωη_h + ... + ω^k-1η_h(k-1))^k

(4) Then (η₀ + ωη_h + ... + ω^k-1η_h(k-1))ⁿ⁺¹= (η₀ + ωη_h + ... + ω^k-1η_h(k-1))ⁿ(η₀ + ωη_h + ... + ω^k-1η_h(k-1))= (a₀η₀ + ... + a_e-1η_e-1)(η₀ + ωη_h + ω²η_2h + ... + ω^k-1η_h(k-1))

(5) (a₀η₀ + ... + a_e-1η_e-1)(η₀ + ωη_h + ω²η_2h + ... + ω^k-1η_h(k-1)) = (a₀η₀)(η₀ + ωη_h + ω²η_2h + ... + ω^k-1η_h(k-1)) + ... + (a_e-1η_e-1)(η₀ + ωη_h + ω²η_2h + ... + ω^k-1η_h(k-1)) = a₀η₀ω⁰η₀ + ... + a_e-1η_e-1ω^k-1η_h(k-1)

(6) It is clear that each of the resulting terms is in the form of a_iη_iω^jη_j*h so if I can prove that each of these form reduces to a rational expression of ω and η_i that I am done.

(7) a_iη_iω^jη_j*h = (a_iω^j)(η_iη_j*h)

(8) Using Corollary 2.1 above and Theorem 2 here, this gives us that each term reduces to:

(a_iω^j)(b₀η₀ + ... + b_e-1η_e-1) where b_i ∈ Q.

QED

Corollary 3.1:

Let ζ be a p-th period of unity.

Let ef=gh=p-1 where f divides g and k=g/f

Let η_i = ζ_i + ζ_e+i + ζ_2e+i + ... + ζ_e(f-1)+i

Let ω be a k-th root of unity

Then:

(η₀ + ωη_h + ... + ω^k-1η_h(k-1))^k = = a₀η₀ + ... + a_h-1η_h-1 + a_hη_h + ... + a_2h-1η_2h-1 + + ... + + a_h(k-1)η_h(k-1) + ... + a_e-1η_e-1

Proof:

(1) From Lemma 3 above, we have:

(η₀ + ωη_h + ... + ω^k-1η_h(k-1))^k = a₀η₀ + ... + a_e-1η_e-1

where the coefficients a₀, ..., a_e-1 are rational expressions in ω over Q [that is, they are rational expressions in Q(μ_k)]

(2) Since f divides g and ef = gh, it follows that g/f = e/h so that h divides e.

(3) This allows us to divide 0 .. e-1 into:

(η₀ + ωη_h + ... + ω^k-1η_h(k-1))^k = = a₀η₀ + ... + a_h-1η_h-1 + a_hη_h + ... + a_2h-1η_2h-1 + + ... + + a_h(k-1)η_h(k-1) + ... + a_e-1η_e-1

QED

Lemma 4:

Let ζ be a p-th period of unity.

Let ef=gh=p-1 where f divides g and k=g/f

Let η_i = ζ_i + ζ_e+i + ζ_2e+i + ... + ζ_e(f-1)+i
Let ξ_j = ζ_j + ζ_h+j + ζ_2h+j + ... + ζ_h(g-1)+j

Then:

η_i + η_h+i + ... + η_h(k-1)+i = ξ_i for i = 0, ..., h-1

Proof:

(1) η_i = ζ_i + ζ_e+i + ζ_2e+i + ... + ζ_e(f-1)+i [From the definition above]

(2) η_i + η_h+i + ... + η_h(k-1)+i =
= [ ζ_i + ζ_e+i + ζ_2e+i + ... + ζ_e(f-1)+i] + [ ζ_h+i + ζ_h+e+i + ζ_h+2e+i + ... + ζ_h+e(f-1)+i] + ...
+ [ ζ_h(k-1)+i + ζ_h(k-1)+e+i + ζ_h(k-1)+2e+i + ... + ζ_{h(k-1)+e(f-1)+i}] =

= [ ζ_i + ζ_{h + i} + ... + ζ_h(k-1)+i ] + [ ζ_{e + i} + ζ_{h + e + i} + ... + ζ_h(k-1)+e+i] + ...
+ [ζ_e(f-1)+i + ζ_h+e(f-1)+i + ... + ζ_{h(k-1)+e(f-1)+i}]

(3) Since k = g/f and ef=gh, it follows that e = gh/f = hk

(4) This then gives us:

[ ζ_i + ζ_{h + i} + ... + ζ_h(k-1)+i ] + [ ζ_{e + i} + ζ_{h + e + i} + ... + ζ_h(k-1)+e+i] + ...
+ [ζ_e(f-1)+i + ζ_h+e(f-1)+i + ... + ζ_{h(k-1)+e(f-1)+i}]=

[ ζ_i + ζ_{h + i} + ... + ζ_h(k-1)+i ] + [ ζ_{hk + i} + ζ_{h + hk + i} + ... + ζ_h(k-1)+hk+i] + ...
+ [ζ_hk(f-1)+i + ζ_h+hk(f-1)+i + ... + ζ_{h(k-1)+hk(f-1)+i}] =

[ ζ_i + ζ_h+i + ... + ζ_h(k-1) ] + [ ζ_hk+i + ζ_h(k+1)+i + ... + ζ_h(2k-1)+i] + ... + [ζ_{h[kf-k] + i} + ζ_{h[kf-k+1] + i} + ... + ζ_h[kf-1]+i]

(5) Since k = g/f, it follows that kf = g so that we have:

[ ζ_i + ζ_h+i + ... + ζ_h(k-1) ] + [ ζ_hk+i + ζ_h(k+1)+i + ... + ζ_h(2k-1)+i] + ... + [ζ_{h[kf-k] + i} + ζ_{h[kf-k+1] + i} + ... + ζ_h[kf-1]+i] =

[ ζ_i + ζ_h+i + ... + ζ_h(k-1) ] + [ ζ_hk+i + ζ_h(k+1)+i + ... + ζ_h(2k-1)+i] + ... + [ζ_{h[g-k] + i} + ζ_{h[g-k+1] + i} + ... + ζ_h[g-1]+i] = ξ_i

QED

Theorem 5:

Let ζ be a p-th period of unity.

Let ef=gh=p-1 where f divides g and k=g/f

Let η_i = ζ_i + ζ_e+i + ζ_2e+i + ... + ζ_e(f-1)+i
Let ξ_j = ζ_j + ζ_h+j + ζ_2h+j + ... + ζ_h(g-1)+j

Let ω be a k-th root of unity

Let t(ω) = η₀ + ωη_h + ω²η_2h + ... + ω^k-1η_h(k-1)

Then:

t(ω)^k has a rational expression in terms of a₀ξ₀ + ... + a_h-1ξ_h-1 where the coefficients a₀, ..., a_h-1 are rational expressions in ω over Q

Proof:

(1) By Corollary 3.1 above, we have:

t(ω)^k = (η₀ + ωη_h + ω²η_2h + ... + ω^k-1η_h(k-1))^k = = a₀η₀ + ... + a_h-1η_h-1 + a_hη_h + ... + a_2h-1η_2h-1 + + ... + + a_h(k-1)η_h(k-1) + ... + a_e-1η_e-1

where the coefficients a₀, ..., a_e-1 are rational expressions in ω over Q

(2) Apply σ^h to both sides of the equation (see Lemma 7, here) gives us:

(η_h + ωη_2h + ... + ω^k-1η_h(k))^k = (η_h + ωη_2h + ... + ω^k-1η₀)^k = = a₀η_h + ... + a_h-1η_2h-1 + + a_hη_2h + ... + a_2h-1η_3h-1 + + ... + + a_h(k-1)η₀ + ... + a_e-1η_h-1

(5) We note that ω^-1(t(ω)) = η_h + ωη_2h + ... + ω^k-1.

(6) We further note that:

[ω^-1t(ω)]^k = (ω^-1)^kt(ω)^k = (ω^k)^-1t(ω)^k = t(ω)^k

(7) So that it is clear that t(ω)^k = (η_h + ωη_2h + ... + ω^k-1η₀)^k = = a₀η_h + ... + a_h-1η_2h-1 + + a_hη_2h + ... + a_2h-1η_3h-1 + + ... + + a_h(k-1)η₀ + ... + a_e-1η_h-1

(8) Since we can make the same argument for ω²t(ω) with σ^2h applied, ω³t(ω) with σ^3h, ..., ω^k-1t(ω) with σ^h(k-1), we can make the following observation:

kt(ω)^k = (a₀ + ... + a_h(k-1))(η₀ + ... + η_h(k-1)) + + (a₁ + ... + a_h(k-1)+1)(η₁ + ... + η_h(k-1)+1) + + ... + + (a_h-1 + ... + a_e-1)(η_h-1 + ... + η_e-1).

(9) Using Lemma 4 above that η_i + η_h+i + ... + η_h(k-1)+i = ξ_i for i = 0, ..., h-1, it follows that t(ω)^k is rationally expressed in terms of ω and ξ₀, ..., ξ_h-1 as:

t(ω)^k = 1/k((a₀ + ... + a_h(k-1))ξ₀ + ... + (a_h-1 + ... + a_e-1)ξ_h-1)

QED

Corollary 5.1: For every integer n, the n-th roots of unity have expressions by radicals

Proof:

(1) This is clearly true if n=1 or n=2 since the roots of unity are (1) or (1,-1)

(2) We may assume that for every integer k less than n, the k-th roots of unity are expressible by radicals

(3) If n is not prime, then since each of its factors are less than n, we can conclude that n is expressible by radicals (see Theorem 2, here).

(4) So, we can assume that n is prime.

(5) We can then order the n-th roots of unity other than 1 with the aid of a primitive root of n [see Theorem 3, here]

(6) We can then consider the Lagrange resolvent (see Theorem, here):

t(ω) = ζ₀ + ωζ₁ + ... + ω^n-2ζ_n-2

where ω is an (n-1)-st root of unity.

(7) By the induction hypothesis, ω can be expressed by radicals.

(8) By the Theorem 5 above (with k = g = n-1), this shows that t(ω)^n-1 has a rational expression in terms of ω

(9) We can then use Lagrange's formula (see Theorem, here), to get the following formula:



QED

References

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