Abel's Form of a General Solution by Radicals

The first step in Niel Abel's 1824 proof on the insolvability of the general quintic equation is the statement that if a general solution to the quintic exists, it can be assumed to have the following form:



where m is a prime number and R, p₂, ..., p_m-1 are all functions of this same form finitely nested and are functions of the coefficients of the general quintic equation.

Today's content is taken from Peter Pesic's Abel's Proof.

Lemma 1:

If a mathematical expression is expressible by radicals, then it is stateable in the form:

(A₁ + A₂ + ... + A_m)/(B₁ + B₂ + ... + B_n) where each A_i, B_i are expressible by radicals.

Proof:

(1) First, I show that if a mathematical expression is minimally expressible by radicals, then it is expressible by the desired form:

a + b = (a + b)/(1)

a - b = (a - b)/(1)

a * b = (a*b)/(1)

a/b = (a)/(b)

√a = (a^(1/2))/(1)

(2) By the above analysis, we know that it works to at least the minimum number of operations. Let's assume that it works only up to n operations.

(3) To complete the proof, I will show that I can add an additional operation to the above form and maintain the above form. I only need to show that this is true for addition, subtraction, multiplication, division, and radicals:

(±)a + (A₁ + ... + A_m)/(B₁ + ... + B_n) =

= (A₁ + ... + A_m + (±)a[B₁ + ... + B_n])/(B₁ + ... + B_n)

a*(A₁ + ... + A_m)/(B₁ + ... + B_n) =

= (a*A₁ + ... + a*A_m)/(B₁ + ... + B_n)

(1/a)*(A₁ + ... + A_m)/(B₁ + ... + B_n) =

= (A₁ + ... + A_m)/(a*B₁ + ... + a*B_n)

[ (A₁ + ... + A_m)/(B₁ + ... + B_n)]^(1/a) = [(A₁ + ... + A_m)]^(1/a)/[(B₁ + ... + B_n)^(1/a)]


QED

Lemma 2:

If the general quintic equation has a solution which has the form:

y = (A₁)^(c₁/d₁) + (A₂)^(c₂/d₂) + ... + (A_n)^(c_n/d_n)

where each A_i is expressible by radicals

then it can be put into the following form:



where R, p₁, ..., p_m-1 are all functions of this same form finitely nested.

Proof:

(1) By assumption, we can state the solution to the general quintic equation in the following form:

y = (A₁)^(c₁/d₁) + (A₂)^(c₂/d₂) + ... + (A_n)^(c_n/d_n)

where each A_i are expressible as radicals.

(2) Now, if we set each A'_i to (A_i^c_i), then we get:

y = (A'₁)^(1/d₁) + (A'₂)^(1/d₂) + ... + (A'_n)^(1/d_n)

(3) Since, the expression has finite complexity, it follows that there can only be a finite number of expressions of the form (A'_i)^(1/d_i)

(4) So, there exists a number k such that:

[number of radicals in y] = k.

(5) Now, we can put this expression in the desired form by setting:

R = A'₁

m = d₁

p₁ = 1

p = (A'₂)^(1/d₂) + ... + (A'_n)^(1/d_n)

p_i = 0 where i ≠ 1.

(6) This changes y into

y = p + p₁R^(1/m)

(7) Now, it is clear that:

[number of radicals in p] + [number of radicals in R] = [number of radicals in y] - 1 = k-1

(8) And since we do the same refactoring for all in radicals in p and all radicals in R and since there are only a finite amount of them, we can put all radicals into the desired form.

(9) To complete the proof, we need only show that an equation without radicals can also be put into this form.

(10) So assume y does not have radicals. That is:

[number of radicals in y] = 0

(13) Then let:

p = A₁^c₁ + A₂^c₂ + ... + A_n^c_n

with each p_i=0.

QED

Corollary 2.1:

If the general quintic equation has a solution, then it can be put into the following form:



where R, p₁, ..., p_m-1 are all functions of this same form finitely nested.

Proof:

(1) Using Lemma 1 above, if the solution to the general equation is expressible by radicals, then it follows that y is expressible as:

y = (A₁ + A₂ + ... + A_m)/(B₁ + B₂ + ... + B_n) where each A_i, B_i are expressible by radicals.

(2) Let:

T = A₁ + A₂ + ... + A_m

W = B₁ + B₂ + ... + B_n

such that we have:

y = T/W

(3) Using Lemma 2 above, we know that T,W can be put in the following forms:





(4) Let:

W(x) = w₀ + w₁x + w₂x² + ... + w_vx^v

α = an n-th root of unity. (that is, α ≠ 1 and αⁿ=1)

(5) So, we now define W₀, W₁, ..., W_n-1 with:

W₀ = W(R^(1/n)) = W

W₁ = W(αR^(1/n))

W₂ = W(α²R^(1/n))

...

W_n-1 = W(α^n-1R^(1/n))

(6) So, if we multiply (W₁*...*W_n-1)/(W₁*...*W_n-1), we get:

Y = (T*W₁*...*W_v-1)/(W*W₁*...*W_v-1)

(7) Multiplying W*W₁*...*W_n-1 out, we get:

W*W₁*W₂*...*W_n-1 =

= (w₀ + w₁R^(1/n) + w₂R^(2/n) + ... + w_vR^(v/n))*(w₀ + αw₁R^(1/n) + α²w₂R^(2/n) + ... + α^vw_vR^(v/n))*(w₀ + α²w₁R^(1/n) + α⁴w₂R^(2/n) + ... + α^2vw_vR^(v/n))*...*(w₀ + α^(n-1)w₁R^(1/n) + α^2(n-1)w₂R^(2/n) + ... + α^v(n-1)w_vR^(v/n)) =

=w₀ⁿ + αw₀^n-1w₁(1 + α + α² + ... + α^n-1) + ... α²w₀^n-2w₂R^(2/n)(1 + α + α² + ... + α^n-1) + ...

(8) Now, since (1 + α + α² + ... + α^n-1) = 0 [See Lemma 2, here], we have:

W*W₁*W₂*...*W_n-1 = v₀ⁿ

(9) If w₀ⁿ contains radicals, then we set W=w₀ⁿ and we repeat the process.

(10) Since the original W has only a finite nesting of radicals, it follows that eventually we will reach a w₀ⁿ that does not contain any radicals and we set each of the resulting t_i = t_i/(w₀ⁿ)

QED

Lemma 3:

For the following expression



We can assume that m is less than n.

Proof:

(1) Assume that m is not less than n.

(2) Then, we there exists q,r such that (see Theorem 1, here):

m = qn + r

where q ≥ 1, and 0 ≤ r ≤ n-1

(3) Then:



(4) Now, if we define p'_i so that:

p'_i = p_i*R^q, then we are left with:



where m is less than n.

QED

Lemma 4:

For any equation of the following form:



We can assume that m is prime.

Proof:

(1) Assume that m is not prime.

(2) Then m consists of a finite number of primes. [See Theorem 3, here for proof of the Fundamental Theorem of Arithmetic]

(3) Let's take out the first prime which I will call f. So that we have m = f*m'.

(4) So we can restate y as:



(5) Now, we can redefine each R such that:

R = R^(1/m')

(6) After this, we are left with an equation of the desired form.

QED

Theorem 5: Abel's General Form

If a general solution to the quintic exists, it can be assumed to have the following form:



where m is a prime number and R, p₁, ..., p_m-1 are all functions of this same form finitely nested and are functions of the coefficients of the general quintic equation.

Proof:

(1) By Corollary 2.1 above:

If the general quintic equation has a solution, then it can be put into the following form:



where R, p₁, ..., p_m-1 are all functions of this same form finitely nested.

(2) By Lemma 3 above, we can assume that m is less than n.

(3) By Lemma 4 above, we can assume that n is prime.

(4) Let R' = R/[p₁ⁿ]

(5) Then we have:


(6) Now, for all this nesting of equations of this form, at the end, each R' must be in a function of the coefficients of the general quintic equation.

(7) If not, then it does not represent a solution to the general quintic since a solution consists of determining each of the roots based on the coefficients given.

QED

References

• Peter Pesic, Abel's Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability, Appendix B, The MIT Press, 2004.