Why There's 'No' Quintic Formula (proof without Galois theory)

Name: Why There's 'No' Quintic Formula (proof without Galois theory)
Uploaded: 2021-07-05T00:51:51.000Z
Duration: 1 h 30 min 4 s

Solving Polynomials Beyond the Quadratic Formula

In this video, we learn about solving polynomial equations beyond the quadratic formula. We explore the cubic and quartic formulas before discussing why there is no algebraic quintic formula. We also introduce complex numbers and De Moivre's theorem.

The Cubic and Quartic Formulas

The cubic formula is similar to the quadratic formula but longer.

The quartic formula is even longer but follows the same pattern as the previous formulas.

These formulas involve addition, subtraction, multiplication, division, square roots, and cube roots.

No Algebraic Quintic Formula

There is no algebraic quintic formula for solving general quintic equations with arbitrary coefficients using only addition, subtraction, multiplication, division, and n-th roots.

Vladimir Arnold's argument explains why such a formula does not exist.

Complex Numbers

Complex numbers can be represented in polar form using magnitude (distance from origin) and phase (angle with real axis).

Multiplying complex numbers in polar form involves multiplying magnitudes and adding phases.

De Moivre's theorem states that raising a complex number to a power involves multiplying its magnitude by that power and adding its phase multiplied by that power.

Fundamental Theorem of Algebra

The fundamental theorem of algebra states that an nth order polynomial equation has n solutions (including repeated roots).

Solutions may be complex.

The Importance of the Term a*z^n

In this section, the speaker explains that the term a*z^n is the most important term in an nth order polynomial and why it is crucial to consider when approximating solutions.

The Significance of a*z^n

The term a*z^n is what makes an nth order polynomial different from an n-1th order polynomial.

When thinking about the complex plane, this term grows most rapidly as z gets bigger. Therefore, it's essential to look at large values of z when approximating solutions.

Moving z around a very large circle allows us to make use of the whole complex plane while considering where z has a large magnitude so that this term dominates.

Shrinking Z Around A Circle

In this section, the speaker discusses how shrinking Z around a circle can help approximate solutions.

Approximating Solutions by Shrinking Z Around A Circle

Shrinking Z around a circle allows us to focus on where Z has a large magnitude so that the term a*z^n dominates.

As we move Z around in circles, its phase changes. For example, if we take n = 2 and have z^2, its phase will be twice as large as that of z. Therefore, it will make two complete circles around the origin for every one circle that z makes.

As we shrink down the path that Z follows, we eventually come across zeros or solutions to our polynomial equation. This is because, as we shrink the path, it must intersect the origin eventually.

Proving the Fundamental Theorem of Algebra

In this section, the speaker explains how to prove the fundamental theorem of algebra using polynomial factorization.

Proving the Fundamental Theorem of Algebra

Starting with an nth order polynomial and finding one solution allows us to factorize that solution out. This gives us an n-1th order polynomial.

We can repeat this process until we get to a factorized form of our original polynomial. This proves that there are n solutions to our nth order polynomial.

Dividing through by the leading term a in our polynomial equation rescales the problem from a set of coefficients to a set of solutions.

Going from Solutions to Coefficients and Back

In this section, the speaker discusses how it is easy to go from solutions to coefficients in a polynomial equation but harder to go back. The speaker explains that there isn't a natural way of ordering the solutions to a polynomial equation.

Going from Solutions to Coefficients

It is easy to go from solutions to coefficients by expanding the brackets.

Swapping two of the solutions won't change the polynomial expression, but it will switch around the set of solutions.

There isn't a natural way of ordering the solutions in a polynomial equation.

Going from Coefficients to Solutions

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Solving Quadratic Equations

In this section, the speaker uses quadratic equations as an example to explain why there isn't a natural way of ordering the solutions in a polynomial equation.

Solving Quadratic Equations

The quadratic formula can be used to find z1 and z2 in terms of w for any given quadratic equation.

By continuously changing the phase of w, we can change the solutions while changing their phase at half the rate that we changed w's phase.

By going around the origin, we can trick someone into reordering their set of solutions.

This property means that the quadratic formula cannot be a proper continuous function because it's multi-valued in terms of w.

Multi-Valued Solutions of Polynomial Equations

In this section, the speaker explains that the quadratic formula involves square roots and that algebraic expressions are limited in how multi-valued they can be.

Multi-Valued Solutions of Polynomial Equations

The quadratic formula has a square root in it, which makes it multi-valued.

Algebraic expressions are somehow limited in how multi-valued they can be.

If a polynomial has solutions that are more multi-valued than any algebraic expression, then it cannot be solved using a formula that only involves addition, subtraction, multiplication, and division mixed up with the coefficient w.

Understanding Roots Better

In this section, the speaker talks about the nature of multivaluedness of nth roots and how they can be used to understand roots better.

Visualizing Fourth Roots of Two

The fourth roots of two have the same magnitude, which is about 1.19.

There are three other choices for the fourth root: negative of that one, i times that, and minus i times that.

These are the fourth roots of unity multiplied by 1.19.

They have phases zero quarters one quarter two quarters and three quarters of a full circle.

Smoothly Changing Quantity

As we smoothly change the quantity we're taking the fourth root of, each root's phase increases smoothly but now at a quarter rate.

If we move around the origin once, we end up rotating four solutions one place around.

If we go around again, they rotate one place further.

As we go around in opposite direction, they rotate back in another direction.

Using Multi-valuedness Property

We can predict how any nth root changes just by asking how many times the quantity inside it goes around the origin.

Whenever choosing paths imagine keeping everything inside roots non-zero.

Biggest Idea in Video: Solving Equations with Algebraic Expressions

In this section, the speaker discusses solving equations with algebraic expressions using quadratic equations as an example.

Solving Quadratic Equations

A quadratic equation cannot be solved with just a rational function of its coefficients because it would be a continuous and single valued function of those coefficients without involving any roots.

But if there was such an expression then it must end up swapping to another root when solutions are swapped around.

This leads to contradiction since z1 is a single valued function of all coefficients and must have gotten back to where it started.

Solving Quintic Equations

We can try to prove that we can't solve a quintic equation using an algebraic expression containing just a single root of b c d e and f.

We imagine z1 as such a function and then ask what properties it has.

It is still continuous but multi-valued as a function of these coefficients.

We need to guarantee that our formula for z1 hasn't changed in order to get the contradiction.

Permutations and Commutators

In this section, the speaker discusses permutations and commutators in relation to nth roots of coefficients. The goal is to find a way to switch around solutions while leaving the nth root unchanged.

Swapping Points

To switch around solutions while leaving the nth root unchanged, we need to swap two points z1 and z2 in some smooth way.

As we swap these points, the quantity r that we're taking a root of will change in some way. It may go around the origin and pick up some total phase.

We can undo the permutation by following the same path in reverse. This will cause r to follow the same path also in reverse, picking up a phase which is just minus 2 pi m.

The total phase change of r is zero, so the phase change of the nth roots is also zero. Therefore, an nth root of r does not actually change at the end of the day.

Commutators

A commutator involves doing one permutation (sigma), then another permutation (tau), undoing sigma, and finally undoing tau.

Doing a commutator between sigma and tau switches around the set of solutions even though every permutation has been done and undone cleverly.

The total phase change of r for a commutator is zero. Therefore, an nth root of r does not change either.

Commutator moves are useful for solving Rubik's cubes because they involve rotating faces and undoing those rotations.

Algebraic Formulas and Nested Roots

In this section, the speaker discusses how formulas that contain only basic algebraic operations and a single nth root can exist. They explain how nested roots are not ruled out yet, unlike adding, subtracting, multiplying or dividing roots.

Ruling Out Non-Nested Roots

The cubic formula involves complicated expressions with nested roots.

Adding together, multiplying or dividing roots is ruled out because each root can be individually made to get back to where it started.

However, nested roots evade this argument because an nth root of coefficients may not necessarily leave the inner nth root unchanged.

Commutator Moves

A commutator move guarantees that an nth root of an expression involving coefficients gets back to where it started.

This works for the inner root but not for the outer root since it may be multi-valued in terms of the inner root.

Following a commutator of commutators leaves expressions involving once-nested roots unchanged.

Proving Triviality of Commutators

To prove there isn't a formula involving nested roots for quintics, we need to rule out nested roots.

All commutators of three points are a cyclic permutation.

Possible three-point permutations result in cyclic permutations when doing one commutator after another.

The Cubic Formula and Group Theory

In this section, the speaker discusses how the cubic formula can be expressed using nested roots and how group theory can help understand this concept.

Commutators of Three Points

The set of three points in the cubic formula remains unchanged.

This is why there can be a cubic formula with once nested roots.

The derived series of the group of permutations of three elements involves commutators.

The first element in the derived series is the group of cyclic permutations c3.

Quartic Formula

There is a quartic formula that has roots of roots of roots inside it.

There are challenges to find commutators that mix up four points in some non-trivial way.

There's no commutator of commutator of commutators which mixes up four points, allowing for a quartic formula to exist with only one extra level of nesting.

Quintic Formula and Commutators

In this section, the speaker discusses how quintic formulas cannot be expressed using any small finite number of nested roots and how commutators can help prove this result.

Commutators for Five Points

If there are commentators of commutators that genuinely mix up switch around a set of five solutions, then there is no way to express its solution using any small print finite number of nested roots.

There are commentators that do mix up five points, ruling out such a quintic formula.

It's possible to obtain a permutation as a commutator by switching around sets cyclically.

Building Arbitrarily Nested Commutators

Cyclical permutations can be written as commentators.

By building arbitrarily nested commutators, we can prove that quintic formulas cannot be expressed using any small finite number of nested roots.

Finding a Representation of Cyclic Permutations as Commutators

In this section, the speaker explains how to find a representation of a one two three cyclic permutation as an arbitrarily nested commutator. This allows for mixing up the set of solutions while leaving an arbitrarily nested root completely untouched and therefore there is no quintet formula.

Representing Cyclic Permutations as Commutators

A way of finding a representation of a one two three cyclic permutation as an arbitrarily nested commutator

Mixing up the set of solutions while leaving an arbitrarily nested root completely untouched

There is no quintet formula

The Beauty of the Argument

In this section, the speaker praises the argument presented in the previous section. They explain that they used to think Galois theory was necessary to prove this result but this approach involving complex numbers and basic properties is much more satisfying.

The Elegance of the Proof

The speaker thinks that it's a beautiful argument

They used to think Galois theory was necessary to prove this result

This approach involving complex numbers and basic properties is much more satisfying

Comparing with Galois Theory

In this section, the speaker compares their approach with Galois theory. They explain that although their approach may be less powerful than Galois theory in some ways, it has its advantages too.

Comparing Approaches

Their approach involves multi-valuedness and continuity which is somehow much more satisfying than using Galois theory

Although their approach may be less powerful than Galois theory in some ways, it has its advantages too

Their approach rules out adding any exponential for example into this, you can't have a quintic formula that involves nth roots of rational numbers

The Value of the Approach

In this section, the speaker explains why their approach is valuable. They discuss how it leads to developing advanced group theory ideas in a satisfying way.

The Value of the Approach

Their approach is valuable because it tells you why you should be thinking about commutators and what's special about them

It leads on to develop a bunch of the ideas that you study in more advanced group theory in quite a satisfying way

It really tells you why you should be thinking about commutators and what's interesting about them

Solvable Permutation Groups

In this section, the speaker discusses solvable permutation groups. They explain that symmetric groups s1 s2 s3 and s4 are all solvable but s5 is not.

Solvable Permutation Groups

The symmetric groups s1 s2 s3 and s4 are all solvable which means that taking commutators of commutators of combinations of communities eventually only gets trivial operations from doing this

This is not true for s5 as there's a whole bunch of different permutations of five points which can be written as arbitrarily nested commutators

You get stuck right when taking conversations of communicators but there's a whole bunch of different permutations of five points which can be written as arbitrarily nested commutators

Perfect Group

In this section, the speaker discusses perfect groups. They explain that the things which can be obtained using their approach are called a5 and it's a perfect group.

Perfect Group

The things which can be obtained using their approach are called a5

A5 is a perfect group, meaning that it's its own commutator subgroup

Anything within A5 can be realized as a product of commentators of things in that group

Conclusion

In this section, the speaker concludes by encouraging viewers to check out the video and paper they learned from. They also ask for feedback on any mistakes or clarifications needed.

Final Thoughts

The speaker encourages viewers to check out the video and paper they learned from

They ask for feedback on any mistakes or clarifications needed

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