Solving Van Roomen's Problem: So far

So, up to know, I've shown that Van Roomen's monster equation: (see here):

(45)x - (3,795)x³ + (95,634)x⁵ - (1,138,500)x⁷ + (7,811,375)x⁹ - (34,512,075)x¹¹ + (105,306,075)x¹³ - (232,676,280)x¹⁵ + (384,942,375)x¹⁷ - (488,494,125)x¹⁹ + (483,841,800)x²¹ - (378,658,800)x²³ + (236,030,652)x²⁵ - (117,679,100)x²⁷ + (46,955,700)x²⁹ - (14,945,040)x³¹ + (3,764,565)x³³ - (740,259)x³⁵ + (111,150)x³⁷ - (12,300)x³⁹ + (945)x⁴¹ - (45)x⁴³ + x⁴⁵ = A

simplifies to this one when n = 45 (see here for proof):



If we generalize the above equation to F_n (see below), there is an interesting recurrence relation that emerges (see here for details).

If we define:

F_n(x) =



Then:

F_n(x) = x*F_n-1(x) - F_n-2(x).

Finally, we can introduce trigonometry to the equation defined above (see here for proof) to get:

for any integer n ≥ 1:

2cos(nα) = F_n(2cosα)

For any odd integer n ≥ 1:

2 sin(nα) = (-1)^(n-1)/2*F_n(2sinα)


Now, let's see if we can simplify the examples that Van Roomen gives for x and A (see here for details). It turns out that all can be simplified using trigonometric identities.

Here are three examples that Van Roomen provides. I should note that in Van Roomen's original problem, he made a mistake on example 2. For purposes of this blog, I have corrected his mistake. (See Jean-Pierre Tignol's book for more details).

Example 1:

If



Then:



Example 2:

If:



Then:



Example 3:

If:



Then:




In my next blog, I will show how each of these values can be simplified using trigonometry.

References

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

Solving Van Roomen's Problem: Step Three

The third step in solving Van Roomen's problem is realizing its relation to trigonometry. Using F_n(x) defined in my last blog (see here), I will show that:

2cos(nα) = F_n(2cosα) where n ≥ 1.

and

2sin(nα) = (-1)^(n-1)/2F_n(2sinα) where n is odd and n ≥ 1.

Here are the details:

Lemma 1: n ≥ 1 → 2 cos(n)α = (2cos α)(2 cos (n-1)α) - 2 cos(n-1)α

Proof:

(1) From the addition and subtraction cosine formulas [see Theorem 1, here], we know that:

cos (a + b) = cos a cos b + sin a sin b

cos (a - b) = cos a cos b - sin a sin b

(2) Adding these two identities together gives us:

cos (a + b) + cos (a - b) = cos a cos b + cos a cos b = 2cos a cos b

Or in other words:

cos (a + b) = 2 cos a cos b - cos (a - b)

(3) Since a,b can be any value, let b = α, let a = (n-1)α

So that:

a + b = (n-1)b + b = (n-1+1)b = nα

a - b = (n-1)b - b = (n-2)α

(4) This then gives us that:

cos (nα) = (cos α)(2 cos(n-1)α) - cos(n-2)α

Or equivalently:

2 cos (nα) = (2 cos α)(2 cos(n-1)α) - 2 cos(n-2)α

QED

This trigonometric identity is relevant to Van Roomen's problem because it fits the same structure as F_n

In my previous blog, I showed that F_n(x) = x*F_n-1(x) - F_n-2(x).

Now, if x = 2 cos α, then we have:

F_n(2 cos α) = (2 cos α)*F_n-1(2 cos α) - F_n-2(2 cos α)

This brings us to the corollary:

Corollary 1.1: 2 cos(nα) = F_n(2 cos α)

Proof:

(1) At n = 1:

F₁ = x = 2 cos α = 2 cos(n α)

(2) At n = 2:

F₂ = x² - 2 = (2 cos α)² - 2 = 2(2 cos²(α) - 1)

Since sin²(x) + cos²(x) = 1 (see Corollary 2, here), we have :

2(2 cos²(α) - 1) = 2(2 cos²(α) - [sin²(α) + cos²(α)] ) = 2[cos²(α) - sin²(α)]

Using the formula for cos(2x) (see Lemma 3, here), we get:

2[cos²(α) - sin²(α)] = 2 (cos 2 α)

(3) So, let's assume that F_n(2 cos α) = 2 cos n α up to n-1 where n ≥ 3.

(4) Now, using our previous formula (see Theorem 1, here), we know that:

F_n(x) = x*F_n-1(x) - F_n-2(x)

(5) Using our assumption in step #3, we have:

F_n(2 cos α) = (2 cos α)*(2 cos(n-1)α) - (2 cos(n-2)α)

(6) Now, using Lemma 1 above, we know that:

2 cos nα = (2 cos α)*(2 cos (n-1)α) - (2 cos(n-2)α) so that by induction (see Theorem, here for review if needed), we have proven that:

F_n(2 cos α) = 2 cos n α

QED

But we are not yet done, we can also show:

Corollary 1.2: For all odd n ≥ 1:

2 sin nα = (-1)^(n-1)/2F_n(2 sin α)

Proof:

(1) From a previous result (see Lemma 1, here), we know that:

cos(z) = sin(z + π/2) where z is any number in radians (see here for review of radians).

(2) Let z = x - π/2

(3) Then:

cos(x - π/2) = sin(x - π/2 + π2) = sin(x)

(4) Now from Corollary 1.1 above, we have:

2 cos(nα) = F_n(2 cos α)

(5) Since α can be any value let α = β - π/2

so that:

2 cos(n[β - π/2]) = 2 cos(nβ - nπ/2) = 2 cos([nβ - (n-1)π/2] - π/2) = 2 sin(nβ - (n-1)π/2)

and

F_n(2 cos (β - π/2)) = F_n(2sin(β))

(6) Using the formula for sin(a+b) [see Theorem 1, here], we get:

sin(nβ - (n-1)π/2) = sin(nβ)cos([n-1]π/2) - cos(nβ)sin([n-1]π/2)

(7) Since n is odd, we know that n-1 is even and there exists m such that (n-1)=2m which gives us:

cos([n-1]π/2) = cos(mπ) = (-1)^m since:

(a) n ≥ 1 so m ≥ 0.

(b) cos(0) = 1, cos(π)= -1 [See Property 6, here]

(c) Finally cos(x + 2π) = x [See Property 5, here]

(d) If m is even, then mπ is divisible by 2π and cos(mπ) = 1 = (-1)^m

(e) If m is odd, then mπ is not divisible by 2π and cos(mπ) = -1 = (-1)^m

Likewise:

sin([n-1]π/2) = sin(mπ) = 0 since:

(a) n ≥ 1 so m ≥ 0.

(b) sin(0) = 0, sin(π) = 0 [see Property 1, here]

(c) sin(x + 2π) = sin x [See Property 5, here]

(d) Putting this together, we can see that mπ ≡ 0 or ≡ π (mod 2π) and either way sin(mπ)=0.

(8) So that we have:

2 sin(nβ - (n-1)π/2) = 2*[sin(nβ)cos([n-1]π/2) - cos(nβ)sin([n-1]π/2)] = 2*(-1)^(n-1)/2*sin(nβ) - cos(nβ)*0 = 2*(-1)^(n-1)/2sin(nβ)

(9) Using step #8 and combining it with step #4 and step #5 gives:

2*(-1)^(n-1)/2sin(nβ) = F_n(2sin(β))

(10) Multiplying both sides by (-1)^(n-1)/2 gives us:

2sin(nβ) = (-1)^(n-1)/2F_n(2sin(β))

QED

References

• "Cos of multiples of x, in terms of cos x ", MathHelp
• Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, 2001

Solving Van Roomen's Problem: Step Two

In my previous blog, I showed how Van Roomen's problem could be reduced to a summation formula:

(45)x - (3,795)x³ + (95,634)x⁵ - (1,138,500)x⁷ + (7,811,375)x⁹ - (34,512,075)x¹¹ + (105,306,075)x¹³ - (232,676,280)x¹⁵ + (384,942,375)x¹⁷ - (488,494,125)x¹⁹ + (483,841,800)x²¹ - (378,658,800)x²³ + (236,030,652)x²⁵ - (117,679,100)x²⁷ + (46,955,700)x²⁹ - (14,945,040)x³¹ + (3,764,565)x³³ - (740,259)x³⁵ + (111,150)x³⁷ - (12,300)x³⁹ + (945)x⁴¹ - (45)x⁴³ + x⁴⁵ =



where n = 45.

In today's blog, I will show how a recurrence relation can be introduced to simplify the problem further.

The key insight comes to generalizing the above summation formula. For example, let's define:

F_n(x) =



It turns out that it is possible to show that for any n ≥ 1:

2cos(nα) = F_n(2cosα)

For any odd n ≥ 1:

2 sin(nα) = (-1)^(n-1)/2*F_n(2sinα)

François Viète knew both of these identities and he would use them to find his solutions to Van Roomen's problem. In today's blog, I will show the proof for the recurrent relation and in my next blog, I will show how this recurrence relation leads to the trigonometric identities above.

Theorem 1: Recurrence Formula

If:

F_n(x) =



Then:

F_n(x) = x*F_n-1(x) + F_n-2(x)

Proof:

(1) F₁(x) = (-1)⁰*(1/1)*[(1!)/(0!1!)]x^1-0 = x

(2) F₂(x) = (-1)⁰*(2/2)*[(2!)/(0!2!)]x^2-0 + (-1)¹*(2/1)*[(1!)/(1!0!)]x^2-2 = x² - 2

(3) F₃(x) = (-1)⁰*(3/3)*[(3!)/(0!3!)]x^3-0 + (-1)¹*(3/2)*[(2!)/(1!1!)]x^3-2 = x³ - 3x

(4) We can see that x*F₂(x) - F₁(x) = x*(x² - 2) - x = x³ - 3x.

(5) So, we can assume that the recurrence formula is true up to some integer n-1 where n ≥ 4, that is:

F_n-1(x) = x*F_n-2(x) - F_n-3(x)

(6) Now, x*F_n-1(x) =



=


(7) Likewise, F_n-2(x) =



=



=



(8) So, we see that: x*F_n-1(x) - F_n-2(x) =





where if n is odd,

C = 0

and if n is even,

C =

-(-1)^{(n/2 - 1)}[(n-2)/(n-1-n/2)][(n-1-n/2)!/(n/2-1)!(n-1-n/2-(n/2-1))!]x^n-2(n/2) =

= (-1)^(n/2)[(n-2)/(n/2-1)][(n/2-1)!/(n/2 -1)!(0!)]x⁰ =

= (-1)^(n/2)[(n-2)]/[(n/2-1)](1)(1) = (-1)^(n/2)(n-2)*(2)/(n-2) =

= (-1)^(n/2)*2

(9) So, x*F_n-1(x) - F_n-2(x) - C - x_n =



(10) Focusing solely on the middle part, we see that:
















(11) If n is odd:

F_n(x) =



(12) If n is even:

F_n(x) =



(13) So, either way we have:

F_n(x) = x*F_n-1(x) - F_n-2(x).

QED

Solving Van Roomen's Problem: Step One

In the last blog, I presented Van Roomen's Problem:

(45)x - (3,795)x³ + (95,634)x⁵ - (1,138,500)x⁷ + (7,811,375)x⁹ - (34,512,075)x¹¹ + (105,306,075)x¹³ - (232,676,280)x¹⁵ + (384,942,375)x¹⁷ - (488,494,125)x¹⁹ + (483,841,800)x²¹ - (378,658,800)x²³ + (236,030,652)x²⁵ - (117,679,100)x²⁷ + (46,955,700)x²⁹ - (14,945,040)x³¹ + (3,764,565)x³³ - (740,259)x³⁵ + (111,150)x³⁷ - (12,300)x³⁹ + (945)x⁴¹ - (45)x⁴³ + x⁴⁵ = A

where:




Each of the coefficients is large and none of them are prime. Nineteen of them, for example, are divisible by 5. 95,634 is divisible by 7. 236,030,652 is divisible by 12. 740,259 is divisible by 3.

In addition, if we examine the coefficients in series from x¹ thru x⁴⁵, we can see that they are smallest at the edges and progressively larger toward the middle. This is a pattern also exhibited by binomials such as (x + y)ⁿ.

François Viète was able to reformulate's Adriaan van Roomen's equation into a summation equation. Using modern notation, he reduced the problem to:



where:

n = 45

= the largest integer ≤ n/2



[See here for review of factorial (!) if needed]


The floor(n/2) function is used because we only want to generate 22 different terms. Van Roomen's equation consists solely of odd powers.

If you want to see that it works, you can work it out for each term. Here are some examples of how the summation formula works:

At i=0:

(-1)⁰*(45/45)*[(45)!/(0!45!)]x⁴⁵ = x⁴⁵

At i=1:

(-1)¹*(45/44)[(44)!/(1!43!)]x⁴³ = -45x⁴³

Let's skip to i=11:

(-1)¹¹*(45/34)*[(34)!/(11!23!)]x²³ = -(45*33!)/(11!23!)x²³ = -(378,658,800)x²³

and so on up to i=22:

(-1)²²*(45/23)*[(23)!/(22!1!)]x¹ = 45x

In my next blog, I will show how a recurrence relation can further simplify the problem.

References

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

Van Roomen's Problem

Adriaan Van Roomen was a professor of mathematics and medicine in Louvain who had developed a strong reputation as one of the top mathematicians of his age. In 1593, he published a survey of the most important living mathematicians. In this work, he proposed a math problem that he dared any of these top mathematicians to solve:

(45)x - (3,795)x³ + (95,634)x⁵ - (1,138,500)x⁷ + (7,811,375)x⁹ - (34,512,075)x¹¹ + (105,306,075)x¹³ - (232,676,280)x¹⁵ + (384,942,375)x¹⁷ - (488,494,125)x¹⁹ + (483,841,800)x²¹ - (378,658,800)x²³ + (236,030,652)x²⁵ - (117,679,100)x²⁷ + (46,955,700)x²⁹ - (14,945,040)x³¹ + (3,764,565)x³³ - (740,259)x³⁵ + (111,150)x³⁷ - (12,300)x³⁹ + (945)x⁴¹ - (45)x⁴³ + x⁴⁵ = A

where:



To show that Van Roomen had a solid understanding of this monstrous equation, he observed the following:

(a) If:



Then:



(b) If:



Then:



(c) If:



Then:



This problem was mentioned in a meeting between the Dutch ambassador and the King Henry IV of France. The Dutch ambassador had noted that not a single French mathematician had been listed in van Roomen's survey of the great mathematicians.

King Henry IV presented the problem to François Viète who was able to solve it. Viète's solution is historically significant because he not only demonstrated how trigonometry can be used to solve algebraic equations, he also showed that this problem possessed 45 different solutions. This result was important in helping to establish the Fundamental Theorem of Algebra.

How did Viète solve it? I will show his solution in my next blog.

References

• Eli Maor, Trigonometric Delights, Princeton Paperbacks, 1998
• Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, 2001
  

François Viète

François Viète was born in 1540 in France; he was a successful politician who also made very important contributions to algebra. He is credited for example with introducing letters to represent known and unknown values in mathematical equations. His writings were also important in establishing the symbol (+) as representing addition and (-) as representing subtraction.

His family was well connected; his mother, for example, was the first cousin of the president of Parliament in Paris. Viète was able to take advantage of these connections through out his life. He attended the University of Poitiers and in 1560, he graduated with a law degree.

Although a lawyer by profession, he took a strong interest in math and science and published his first mathematical paper in 1571. He was especially interested in the works of Pappus and Diophantus.

Viète was a Huguenot and he lived in Paris during the St. Bartholomew's Day Massacre where many thousands of protestant Hugeuenots were killed.

In 1573, he became a councillor at Rennes where he remained until 1580. At that time, he was appointed by King Henry III to be a royal privy councillor in Paris. In 1584, King Henry III's younger brother became ill and died; this meant that Henry of Navarre, a protestant, became heir to the throne. In the struggle that erupted, Viète, a protestant, was kicked out of office.

Viète left Paris and headed to the small town of Beauvoir-sur-Mer. He spent five years there where he was able to devote time to studying mathematics. It was during this time that he did most of his most imporant work relating to cubic equations and mathematical notation.

Viète believed that the Greeks had not revealed all their mathematical insights. He believed that they had secret methods which he hoped to rediscover. In this way, he introduced what would later become variables and coefficients. In Viète's view, he had restored the hidden mathematical methods of the Greeks.

In 1589, he was called back to parliament in Tours. This happened just before Henry III was assassinated on August 1, 1589. He stayed on as the protestant Henry of Navarre became King Henry IV.

In 1590, a coded letter to King Philip II of Spain was intercepted by the French. Viète, at this time, had a very strong reputation in mathematics and he was given the task of deciphering the note. This he accomplished in March of 1590. The MacTutor web site quotes a historical text:

... when Philip, assuming that the cipher could not be broken, discovered that the French were aware of his military plans, he complained to the Pope that black magic was being employed against his country.

When in 1592, King Henry IV converted to Catholicism. Viète did the same.

In 1593, there was a challenge made to all mathematicians by Adriaan van Roumen, a professor of math at Louvain. Roumen asked for a solution to an equation which had 45 terms. Viète was able to solve this problem. In fact, Roumen asked for one solution. Viète was able to demonstrate that there were 23 positive solutions.

Viète was very wealthy. He published all of his writings himself and sent them out to scholars throughout Europe. In all of his writings, he only focuses on positive values. Like many of the mathematicians before him, he did not include consideration of negative numbers in his writings.

He was a member of the Royal Privy Council until his death in February, 1603.

References

• "François Viète", Wikipedia.Com
• "Biography of François Viète", MacTutor Math Site
• Jennifer Orlansky, "François Viète: Father of Modern Algebraic Notation", Rutgers Term Paper
• Grace Tumminelli, "The Lawyer and the Gambler", Rutgers Term Paper
  
• Peter Pesic, Abel's Proof, The MIT Press, 2003.
  

Proof of the Solution to the General Quartic Equation

In today's blog, I provide the proof for the solution of the general quartic equation. If you would like to see how Lodovico Ferrari found his solution, see here.

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

Lemma 1: Depressed Quartic Equation

If:

x⁴ + bx³ + cx² + dx + e = 0

Then there exists y,p,q,r such that:

y⁴ + py² + qy + r = 0

where:

y = x + (b/4)

p = c - 6(b/4)²

q = d - (b/2)c + (b/2)³

r = e - (b/4)d + (b/4)²c - 3(b/4)⁴

Proof:

(1) Let y = x + (b/4) so that x = y - (b/4)

(2) We note that (see here for details on the Binomial Theorem if needed):

(u - v)² = u² - 2uv + v²

(u - v)³ = u³ - 3u²v + 3uv² -v³

(u - v)⁴ = u⁴ - 4u³v + 6u²v² -4uv³ + v⁴

(3) This then gives us:

(y - [b/4])⁴ = y⁴ - by³ + (3/8)b²y² -b³y/16 + (b/4)⁴

b(y - [b/4])³ = by³ - (3/4)b²y² + (3/16)b³y - b⁴/64

c(y - [b/4])² = cy² - bcy/2 + c(b/4)²

d(y - [b/4]) = dy - bd/4

(4) Putting this all together gives us:

x⁴ + bx³ + cx² + dx + e =

= y⁴ + (3/8)b²y² -b³y/16 + (b/4)⁴ - (3/4)b²y² + (3/16)b³y - b⁴/64 + cy² - bcy/2 + c(b/4)² + dy - bd/4 +e =

= y⁴ + [c-(3/8)b²]y² + [ (1/8)b³ -bc/2 + d]y + [e - bd/4 + c(b/4)² - 3b⁴/256 ]

QED

Lemma 2: Solution for y⁴ + py² + r = 0

If y⁴ + py² + r = 0

Then:

y = ± √ (1/2)[-p ± √p² - 4r

Proof:

(1) y⁴ + py² + r = 0

(2) (y²)² + p(y²) + r = 0

(3) Using the quadratic equation (see here):

y² = (1/2)[-p ± √p² - 4r

(4) Solving for y², we get:

y = ± √ (1/2)[-p ± √p² - 4r

QED

Lemma 3:

If:

y⁴ + py² + qy + r = 0 and q ≠ 0

Then there exists u ≠ 0 such that:

(y² + p/2 + u)² = [√2uy - q/(2√2u)]²

Proof:

(1) y⁴ + py² + qy + r = 0

(2) y⁴ + py² + (p/2)² = -qy -r + (p/2)²

(3) (y² + p/2)² = -qy -r + (p/2)²

(4) Let u be the solution of the following equation:

8u³ + 8pu² + (2p² -8r)u - q² = 0.

We know that this solution exists from the general cubic equation (see here)

(5) We know that u ≠ 0 since if u = 0, then q = 0 but q ≠ 0.

(6) So, we can rearrange our equation in step #4 to:

8u³ + 8pu² + (2p² -8r)u = q²

And after dividing both sides by 8u, we get:

u² + pu + (p/2)² - r = q²/8u.

(7) Now, we know that:

(y² + p/2 + u)² = -qy -r + (p/2)² + 2uy² + pu + u²

(8) So, using step #6, we have:

-qy -r + (p/2)² + 2uy² + pu + u² =

= -qy + 2uy² + q²/8u =

= [√2uy - q/(2√2u)]²

QED

Thereom: General Quartic Equation

If:

ax⁴ + bx³ + cx² + dx + e = 0

p = (c/a) - 6([b/a]/4)²

q = (d/a) - ([b/a]/2)c + ([b/a]/2)³

r = (e/a) - ([b/a]/4)d + ([b/a]/4)²c - 3([b/a]/4)⁴

s = ([b/a]/4)

u = the solution to the cubic equation: 8u³ + 8pu² + (2p² -8r)u - q² = 0.

Then:

if q = 0,

x = ± √ (1/2)[-p ± √p² - 4r - s

if q ≠ 0, then:

x = ± (1/2) [√2uy + √2u - 2p - 4u + 2q/(√2u)] - s

or

x = ± (1/2) [-√2uy + √2u - 2p - 4u - 2q/(√2u)] - s

Proof:

(1) ax⁴ + bx³ + cx² + dx + e = 0

(2) If we divide all sides by a, we get:

x⁴ + (b/a)x³ + (c/a)x² + (d/a)x + (e/a) = 0

(3) Using Lemma 1 above, we get:

y⁴ + py² + qy + r = 0

where:

y = x + ([b/a]/4)

p = (c/a) - 6([b/a]/4)²

q = (d/a) - ([b/a]/2)c + ([b/a]/2)³

r = (e/a) - ([b/a]/4)d + ([b/a]/4)²c - 3([b/a]/4)⁴

(4) If q=0, then using Lemma 2 above, we have:

y = ± √ (1/2)[-p ± √p² - 4r

(5) If q ≠ 0, then using Lemma 3 above, we have:

(y² + p/2 + u)² = [√2uy - q/(2√2u)]²

where

u is the solution to:

8u³ + 8pu² + (2p² -8r)u - q² = 0.

We can find this solution using the cubic equation (see here).

(6) This then gives us:

y² + p/2 + u = ± √2uy - q/(2√2u).

(7) This then gives us four solutions:

(a) Case I: y² + p/2 + u = + √2uy - q/(2√2u).

In this case:

y² - √2uy + [p/2 + u + q/(2√2u)] = 0

Using the quadratic equation:

y = ± (1/2) [√2uy + √2u - 2p - 4u - 2q/(√2u)]

(b) Case I: y² + p/2 + u = - √2uy + q/(2√2u).

In this case:

y² + √2uy + [p/2 + u - q/(2√2u)] = 0

Using the quadratic equation:

y = ± (1/2) [-√2uy + √2u - 2p - 4u + 2q/(√2u)]

(8) Since x = y - ([b/a]/4), we have the following solutions:

(a) If q=0, then:

x = ± √ (1/2)[-p ± √p² - 4r - ([b/a]/4)

(b) If q ≠ 0, then:

x = ± (1/2) [√2uy + √2u - 2p - 4u - 2q/(√2u)] - ([b/a]/4)

or

x = ± (1/2) [-√2uy + √2u - 2p - 4u + 2q/(√2u)] - ([b/a]/4)

QED

References

• Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, 2002

Ferrari's Solution to the quartic equation

In today's blog, I go through Lodovico Ferrari's solution of the general quartic equation. In my next blog, I will present a proof of Ferrari's method.

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

(1) Ferrari started out with a quadratic equation of the following form:

x⁴ + bx³ + cx² + dx + e = 0

(2) Then, he borrowed a method from Girolamo Cardano (see here) and set y = x + (b/4) which means x = y - (b/4).

(3) Substituting x = y - (b/4) gives us:

(y - [b/4])⁴ + b(y - [b/4])³ + c(y - [b/4])² + d(y - [b/4]]) + e = 0

Working this through (see details here), gives us:

y⁴ + py² + qy + r = 0

where:

p = c - 6(b/4)²

q = d - (b/2)c + (b/2)³

r = e - (b/4)c + (b/4)²c - 3(b/4)⁴

(4) Now if we move the q and r term over we get:

y⁴ + py² = -qy -r

This helps us because if we add (p/2)² to both sides, we can complete the square to get:

(y² + p/2)² = -qy -r + (p/2)²

(5) Now, if we add a value u to our square, we get the following:

(y² + p/2 + u)² = -qy -r + (p/2)² + 2uy² + pu + u²

This may not appear to help us but it gives us a possibility for completing the square on the other side.

(6) For example, if we could find a value u such that:

-qy -r + (p/2)² + 2uy² + pu + u² = ([√2uy - q/(2√2u)]²

Then we have found a solution (see step #11 if this point is not clear)

(7) Now:

([√2uy - q/(2√2u)]² = 2u²y² - 2*(√2uy)(q/(2√2u)) + q²/(8u) =

= 2u²y² - q/u + q²/(8u)

(8) So this equation works only if:

-qy -r + (p/2)² + 2uy² + pu + u² = 2uy² - qy + q²/(8u)

That is if:

-r + (p/2)² + pu + u² = q²/(8u)

(9) But if we multiply 8u to all sides and then subtract by q², we get:

8u³ + 8pu² + (2p² - 8r)u - q² = 0

(10) This is the general cubic equation which we can solve using Cardano's formula so we know that u can take on this value.

(11) This then gives us:

(y² + p/2 + u)² = ([√2uy - q/(2√2u)]².

(12) This can we solve using the quadratic equation. The four solutions can be found from:

(y² + p/2 + u)² = ± [√2uy - q/(2√2u)]

I should probably note that we assumed that u ≠ 0. I will talk about the situation were u = 0 in my next blog where I provide a proof for Ferrari's solution.

References

• Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, 2002