Galois Theory Explained Simply

[Applause] solving a quadratic equation is easy and that's because we have a formula we can obtain the roots from the coefficients of the equation by simply using addition subtraction multiplication division and taking a root if roots of a polynomial equation can be obtained in this way they say that an equation is solvable by radicals and it basically means that there exists a simple way of finding its roots which is by some combination of the operations we just mentioned but are all polynomial equations solvable by radicals does a similar formula exist for equations of high degree today we'll answer this question using one of the most amazing and quite sophisticated methods in mathematics known as gala theory but first let's talk about numbers if we only consider equations with rational numbers as coefficients the roots will quickly get us outside of this basic set so the roots of a simple equation x square minus 2 equals 0 are irrational numbers square root of 2 and minus square root of 2. let's append these two roots to the set of all rational numbers but before we move on let's get back to rational numbers just one more time rational numbers have one amazing property when we add subtract multiply or divide two rational numbers with the only exception of division by zero we still get a rational number in algebraic terminology this means that rational numbers form a field but after appending the two roots of our equation we no longer have a field for instance adding 1 and square root of 2 already gets us outside of our set so instead of just appending the two roots let's also append everything we can get as a result of any legitimate algebraic operations performed any finite number of times and now we got ourselves a field we extended the field of rational numbers with the help of square root of two and minus square root of two but it's easy to see that it was enough to only use square root of two because minus square root of two can be obtained by multiplying square root of 2 by -1 this new field by the way consists of all numbers of the following form where x and y can be any rational numbers similarly for any polynomial equation we can create an extension field of rational numbers with the help of all of its roots but let's get back to our previous example and see what happens if in this extended field we switch the roots around so let's pick some element from our field and in it we will replace square root of 2 with minus square root of 2. similarly instead of this number we would have this one this mapping certainly shuffles numbers around within our field but something important remains unchanged first of all notice that all rational numbers are not affected by this transformation but there is more let's give this mapping a name let's use some fancy greek letter for that phi so we can rewrite our expressions as follows it's very easy to check by simply performing the required arithmetic operations that for any two numbers a and b from our field phi of a plus b equals phi of a plus phi of b in other words phi preserves addition but it also preserves subtraction multiplication and division if we now take an arbitrary number x plus square root of 2 y from our field and look at x and y as coordinates on the plane then applying phi acts as a mirror reflection over the x-axis a kind of a symmetry but what if we applied symmetry phi twice to our field a number would reflect off of the x-axis once and then one more time basically bringing it back to where it was so applying phi twice in a row acts as a trivial transformation which is also a symmetry it only doesn't rearrange anything in our field it just keeps everything in its place technically any two symmetries phi and lambda of a given equation can be applied sequentially and the result will be another symmetry that we denote as follows applying the symmetry and then the other looks like an operation over symmetries we use the little circle there and it looks a lot like some sort of multiplication as a matter of fact the symmetry that does not change anything in the field acts as an identity element among symmetries and is often denoted as epsilon and x precisely like 1 when multiplying numbers also no matter what the symmetry is by performing it backwards we get an inverse symmetry in our prior example of equation x squared minus 2 equals 0 element phi that was switching the two roots is actually inverse to itself because phi multiplied by phi gives us a trivial symmetry epsilon all symmetries of a given equation are called the galwa group of that equation and solvability of an equation by radicals is tightly connected to properties of the equation's galwa group and that's what we are about to uncover further our simple example of equation x square minus 2 equals 0 gave us a two element group that consists of epsilon the trivial symmetry and phi the symmetry that switches the two roots and can be thought of as a mirror reflection or just turning the set by 180 degrees using this analogy we can say that an equation x to the seventh minus two equals zero will have 7 roots and will have gallowa group of 7 elements that can be thought of as rotations of a heptagon and rotating the heptagon by one notch will be an element phi of the group and every other element can be obtained by applying phi multiple times which is the same as powers of phi this equation by the way solved by radicals because all we need is to take seventh root of two and just like in the previous equation we got square root of two and minus square root of two as the solutions here we have seven generally speaking complex numbers of a slightly more complicated form but we don't really care all we care about is that solving the equation requires obtaining a seventh root of two of a number and that is something we know how to do so this is a good gala group good in a sense that the corresponding equation is solvable by radicals groups like this one are called cyclic as you can cycle through all the symmetries in the entire group by applying just one symmetry over and over again but to be good gala group doesn't necessarily have to be cyclic it can be composed of multiple different cyclic groups just like a combination lock is composed of multiple dials with digits so the following equation can be rewritten as follows and we again know how to solve it we just need to solve each of the component equations x to the seventh minus one equals zero and x to the fifth minus one equals zero and we know how to do that by calculating seventh and fifth roots respectively the galac group for this twelfth degree equation will look like a combination lock with two dials with seven and five digits this is no longer a cyclic group but it still has one important property that cyclic groups have for any two symmetries phi and lambda of such a group applying phi and then lambda is the same as applying lambda and then phi wait what now this might sound weird but this is where our so-called multiplication of symmetries differs from multiplication of numbers when multiplying numbers order doesn't matter but with symmetries or more generically with transformations it may here's a little analogy that will help you understand this concept better imagine that we are transforming the look of a person let's say we have two transformations phi putting on a watch and lambda putting on shoes if we apply phi and then lambda this is what we get now let's apply these two transformations in the opposite order we apply lambda and then we apply phi and this is what we get which is exact same result so in this case we say that phi multiplied by lambda is the same as lambda multiplied by phi but let's now consider a slightly different pair of transformations psi putting on underwear and ita putting on trousers applying psy and then eta gives us roughly speaking a very conventional outcome but applying transformation ita and thenxi produces a noticeably different result fyi when multiplication order doesn't matter a group is called a billion named after prominent norwegian mathematician niels abel and guess what when the order does matter we call the group not ibelin so then cyclic groups are not the only good ones a billion groups are also good because an equation with an a billion gallon group is solvable by radicals now our rebellion groups have an interesting property let's pick one such group from the previous example we can say that this group can be achieved by extending one cyclic group say this one with another cyclic group and even if our a billion group was bigger and the corresponding combination log had more dials the group can be achieved as a sequence of similar extensions we start with a cyclic group and extend it with another cyclic one then the result of it is again extended with a cyclic group the result of that extension in turn is again extended with a cyclic group and so on every time we extend with a cyclic group basically attaching a dial at a time to our combination lock but the combination lock approach is not the only way to extend one group with another let's take one dial with six digits on it but this time allow to not only rotate it as before but also flip it like so this group is also an example of extending one cyclic group with another cyclic group more specifically a group with six elements with a group of two elements but the result will be a non-ibelian group because performing a simple rotation and then flipping gives us a different result than flipping first and then rotating and by the way this particular group is a gala group for the following equation every time we can obtain our gala group as a sequence of extensions by cyclic groups the equation will be solvable by radicals and it doesn't matter how we extend it whether using a combination lock approach or by flipping a polygon or even in some completely different way this type of groups was subsequently named solvable to reflect the fact that the corresponding equation is solvable by radicals but are all gala groups solvable in other words can any galwa group be obtained as a series of cyclic extensions the answer is no if you consider this equation its gala group is the group of all possible permutations of the equations 5 roots and is usually denoted as s5 and it turns out that you cannot build this group as a series of cyclic extensions permutations of 2 3 and 4 roots sure 5 and above no and this is the beauty of galwa theory it first establishes that an equation is solvable by radicals if and only if its group of symmetries which we call the gallowa group is solvable this is a military maneuver of taking the battle to the turf where you have a greater chance to win and once you did you deal with groups and not with polynomials anymore and here we can prove for instance that the group of all permutations of five six or more roots is non-solvable and the immediate implication of that is that the corresponding equations are not solvable by radicals either so the general answer is only equations of degree lower than 5 are all solvable by radicals but starting with 5 and above there are equations that do not have a formula in radicals galois theory that rests on the great work of ruffini lagrange abel and then ultimately every state galwa is one of the roots of the broad and powerful branch of mathematics known today as group theory with applications in chemistry material science physics cryptography and other domains the idea that instead of studying some objects directly we may study their symmetries and then from there much easier obtain some properties of the objects themselves is a very powerful idea and it's amazing that galois group was one of the earliest examples of that greatness of human thought thank you for watching if you find this video helpful please hit the like button share it with friends and subscribe to my channel

Channel: Math Visualized

Views: 181,303

Rating: 4.9133739 out of 5

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Id: Ct2fyigNgPY

Length: 14min 44sec (884 seconds)

Published: Mon Nov 09 2020

Reddit Comments

[u/cocompact]

It is so unfortunate that multiple examples in this video are incorrectly described.

1. He says (7:20) the Galois group of x⁷ - 2 (over Q) is cyclic, which is incorrect: it is a dihedral group of order 14. [Correction: the Galois group is the ax+b group mod 7, of order (7)(6) = 42.]

2. He says (8:10) the Galois group of (x⁷-1)(x⁵ - 1) is a product of cyclic groups of orders 7 and 5, but actually it's a product of cyclic groups of order 6 and 4 (neither polynomial factor is irreducible, since each has the root 1). He says the Galois group "is no longer a cyclic group" but if it were a product of groups of relatively prime orders 7 and 5 as he says, then that product of cyclic groups would be a cyclic group.

3. He says (12:40) the Galois group of x⁵ - 2x + 1 is S5, but that polynomial is reducible (1 is a root) and its Galois group is S4, so its roots are solvable by radicals (in fact, by the quartic formula).

This video is a terrible example of educational content since nearly every Galois group it introduces is described incorrectly. The creator of the video should delete it from YouTube and post a new one with those major errors fixed. Imagine this were a video on differential equations where four examples of differential equations are presented and three of them are solved incorrectly. It shouldn't be praised if three out of four examples are wrong in serious ways.

[u/thereforeqed]

This video is excellent, but it explains exactly the parts that I do understand about using Galois theory to prove the insolubility of the quintic and above and glosses over exactly the parts I do not understand.

[u/Miyelsh]

I came across this video on youtube. I've seen quite a few explanations of galois theory, but this one really connected the dots for me on why taking polynomials into the domain of exploring symmetries, allowed for mathematicians to show why no general solutions to quintic of higher polynomials was possible.

[u/mrtaurho]

Saw this video the other day in my YouTube recommendations. Didn't watched it. Oh boy... Thanks for bringing it back to my attention! Definitely worth watching :D

[u/[deleted]]

Nice video! I hope this channel makes more content.

[u/SkinnyJoshPeck]

I do not like that the author kept using literal division operator. It is confusing - defining everything in terms of inverses helps to keep away from talking about "dividing by zero" which is something I hate. Much easier to say "zero doesn't have an inverse. How would you multiply a number by 0 and get 1?" and bang, no reason to talk about dividing by zero, especially if you're going to make a video about Galois Theory. Just say inverse and not division since you introduce them as two separate concepts even though they aren't, and then combine them later anyways, or just start with a better definition of what a field is. While technically it's fine, it's like hearing bad grammar.

[u/Chhatrapati_Shivaji]

I guess that's another maths YouTuber to subscribe

[u/Nonabelian]

I wish I leveraged more math you tubers during undergrad

[u/LifeOfAPancake]

Most of the examples given seem to have mistakes but nonetheless this video was great conceptually. It reminded me why I studied maths to begin with. My courses in abstract algebra rushed through the content so I never quite had the opportunity to sit back and enjoy the beauty of galois theory. I hope to someday go back and spend a few months really taking my time reading through a few good books on it