Finite Fields & Return of The Parker Square - Numberphile

oh my goodness we've got a fresh bit of maths research well from 2019 i consider that very recent all things considered about gaussian integers rings finite fields and the magic square of squares and so uh for some of you that's a bit of a spoiler on where parts of this video may be going some of you may know the parker square this is a fan sent this in if you don't know what the parker square is that's great to hear i don't want to call it the parker square because it doesn't work properly it would be like oh that's a classic parker square we'll get to this content in a moment uh first of all i really want to focus on finite fields so a field is a bit it's like a generalized version of what is a number in terms of you can multiply it you can add it and very importantly you can find its inverse so for example the numbers the real numbers are a field because if you've got any if you've got a number i don't know a and you multiply it by b you're going to get exactly the same thing as if you've got b multiplied by a which means that they're commutative and there will exist some number the inverse of a so if you multiply a by 1 over a then you get 1. so that's a multiplication inverse and so for any number if you had for example f of 4 and then you multiply that by 0.25 then you get 1 and very importantly this is a real number this is a real number and so every real number has a buddy real number where if they multiply together you get 1 which means you can undo multiplication it means you've got division and you're off and racing and you can do this for all sorts of cool things you can do it like i said here for the real numbers you can do it for complex numbers uh you can do like some polynomials if you want to go wild rationals the field is wide open so i'm trying to say except the integers the integers don't work very sadly big fan of whole numbers everything else is great you can add them you can multiply them and if you multiply them because the other thing i should say if you multiply two of them together a times b and you get some number c that is also a real number or it's it's a member of the field so the field's closed there's nothing you can do in the field that will cause the ball to land outside the field anything if you're going to multiply your ad you land somewhere else in the same field very important and you've got this here whereas integers that works you multiply two integers together you land somewhere else in the field of integers but you cannot find an inverse a whole number inverse for an integer so that means they get kicked out unless you shrink the field a bit so instead of having the infinite uh as let's say there's infinitely many integers there's a big infinity of real numbers instead of having too much space we want to we want to narrow this down a little bit if you have a finite field suddenly your integers are back on the table and you can have inverses of whole numbers provided the field is finite you know what let's um very quickly put together the addition and multiplication grids for a finite field across some integers so uh brady do you have a fun prime number and i should warn you the bigger the prime you name the longer this is going to take don't go trivial don't i miss your seven seven that's nice that that's neat and tidy so what we'll do is surface seven we're gonna effectively be doing modular arithmetic on mod 7. and so what we're going to do is let's do multiplication first and i'll very quickly fill in this table so we're going to multiply here we go we're going to multiply as if this is a grid so one times one is one let me this is a super boring column this one's not much more exciting so two times one two two twos are four six ah now this this should be eight mod seven one and so here we should have ten so that's going to be three and here we should have twelve so it's going to be five so i'm taking whatever you get when you multiply these i'm getting the remainder mod seven so i'm dividing out any sevens it's just the leftover bit and we can do the same thing for addition but we add them and we take the mod seven and we loop back now at this point a lot of people thinking well hang on why are you getting so excited about finite fields when you're just doing modular arithmetic that doesn't count well it's not 100 true so i would call this this is like the the integers uh so z weird z for the integers uh kind of mod 7 gives us this this doesn't work for any number you don't get a finite field which is why i said prime it only works for primes and powers of primes so you will get a finite field for the integers mod 7 you'll get a finite field for the integers mod 49 so that's 7 squared or mod 25 because that's 5 squared or whatever the case it doesn't work for the in-between values now i've never tried this but i thought it might be fun just to see what happens when it breaks so let's break it let's brady you want to pick a number which is not a prime and is not a power of a prime and then we'll we'll try to do the same grid and we'll see we'll see why it breaks 6 6 excellent because that's 3 times 2 it's not a power of a prime it's got more than one prime factor is the other way of saying that so so this is going to be fancy zed six okay so i should point out i i've kind of hidden zero because a whole row of zeros is even less exciting than um having these and zero doesn't get inverses and all that jazz so we ignore zero now so you think well this is perfectly good why doesn't this count as a finite field well it's because we've haven't got our inverses so if you ever look over here all the numbers we might have all the non-zeros one through six one has an inverse it's one because there's one here two has an inverse it's four so if you multiply two by four mod seven you get back to one so technically a half in the finite field seven is four so that checks out uh three a third is five because 3 times 5 gives you 1 and then there's one there and there's one there and there's one there and so quite nicely there is a 1 in every single row and there's a one in every single column as you set up here we have to be able to achieve one somehow yes you have to multiply to get to one and you can't do that with regular integers but because you've got mod it means you can loop around to get back to one because otherwise once you once you're gone you're not you're not coming back to one whereas now loops back however sometimes you never hit the one so there's a there's some ones in this grid but look at this 2 hasn't got a inverse there's nothing you can multiply 2 by to get back to 1 mod 6. and so that's why this one is ruled out it's still nice still modular arithmetic big fan it's not a finite field and finite fields then linked to all other amazing things and loads of abstract mathematics you'll be looking at fields and finite fields which are great and finite fields have allowed for a breakthrough in an area of mass very close to my heart when i first read a paper i'm just kind of glancing through it i'm not checking through all the details i just want to get a sense of what they're doing and you know what i do and don't understand and then what really jumped out at me is section five here the smallest non-parker finite field it turns out they're now using my name that's me i'm parker as a type like a a property that finite fields can have in fact you can put all finite fields into two categories you can put them into the parker finite fields and you can put them into the non-parker finite fields and a very good question might be which is better and very sadly a a parker finite field means it doesn't work it's like the dad it's the dud finite fields when they're growing up what do you want to be when you're exactly the same size because you're finite and they say i hope i'm non-parker that would be the dream to be a non-parker finite field so i've got a thing named after me i'm afraid your field has been diagnosed as i'm the phrase parker it's not curable the condition of being parker oh and they've used uh fancy f to mean finite field mod 29 for example is not parker since here and then they've given a magic square so for everyone who's unfamiliar with magic squares of squares this is a really nice entry point so this is a magic square and it's got nine numbers in it and so a magic square if you want it to be properly magic is if you add together all the columns you always get the same total if you add together all the rows you always get the same total and people are really insistent about this it turns out if you add together both of the diagonals you get the same a total and then you can go insane you can do all these like pan diagonals that wrap around but people don't tend to put those in they just care about both of those diagonals that doesn't work for these numbers until we square them you now have a magic square where every number in it is a square number and it works hence magic square of squares here's magic square of squares now this has one extra condition so so the full backstory is i came up with a magic square of squares because i thought it was interesting no one had ever found one and no one had ever managed to prove that one doesn't exist which is still true for the record so i gave it a go i found this one now mine has the extra condition it's not very good and that's because it's got the duplicate numbers so you can in fact it's symmetric which is terrible and it doesn't add it doesn't do all of them one of the diagonals is wrong near miss and this bend up the parker square and i don't know who was responsible for this but a line of merch came out people started showing up to my shows wearing t-shirts very few performers have put up with being bullied by their merch at their own gigs this one has a condition as well so it is it's got both diagonals it's got every row every column except the whole thing only works mod 29 so this works on the finite field of integers mod 29 we can we can check that i i would like to do that and you can decide how much stays in the edit and now these are all mod 29 so anything smaller than 29 doesn't budge anything bigger than 29 so now we need to go through and add these up so we'll do the rows first shall we so 23 and 25 is 28 29 so if we come across there add them up we get 29 this one here we get 29. uh this one here you're thinking whoa it's gonna be bigger than 29 28 and 24 is gonna be 52 58 and you're like that's way too big well actually 58 mod 29 well it's zero in fact all of these mod 29 a zero so actually if you add every single direction you get zero so it's a magic square of squares where the magic sum is zero so that's a beautiful i don't know why i find that so pleasing and look what dross you served us up i died i i was four years early like you can't look like it's like looking at the iphone 4 and going well that wasn't very good yeah things have moved on technology has advanced without this this wouldn't be happening that's right what a weird world we live in brady so 29 like that's a par that's a non-parker field it's a non-parker field because within the numbers in the finite field of order 29 there exists a perfect three by three magic square of squares and so any uh finite field where you can make a magic square of squares is a non-parker finite field because it works and then the rest of the paper of this section of the paper this is really quite nice they then go they go well hang on okay we found this one and then they say uh here's all the finite fields with orders less than 29 and so you've got a whole bunch of them here they're all the primes and powers of primes below 29. so now they found the one for 29 and the obvious question is is that the smallest this is what you always ask in math is it the smallest is it the biggest and so their first question was is 29 the smallest finite field which is non-parker so is everything below that unfortunately resigned to being parker and when you say finite fields we're narrowing this search to the integer finite fields obviously yeah yeah and you could do magic squares with other things check some polynomials in there you could do complex in fact gaussian numbers they're talking about earlier uh when when you're looking at it um in the complex space right so there's a lot of fun you can have here's the deal they go through they rule them all out none of them are non-parker everything below 29 is parker and so they managed to show comprehensively that this is the smallest non-parker finite field but then they turn and look in the other direction in general they suddenly realize actually most finite fields are non-parker because of the parker ones the ones that don't work are quite rare and so here it's a conjecture so this is not proven but they seem pretty confident the only parker fields of prime order are these ones and so it turns out any finite field with a prime order of our friends the integers bigger than 67 is non-parker all of them the infinitely many prime numbers bigger than 67 all have a magic square of squares in there somewhere in their finite field so it turns out not working being a parker is actually quite a rare um special property prestigious exactly bestowed upon very few people and finite fields get to have the luxury of being parker because uh it's conjectured but i i'm i i'm going to believe them because once you go plastics they're all they're all non-parker now the flip side to this is you might say well what about infinite fields like what what about the integers which is what started this whole thing in the first place for the infinite field of um integers well this is no longer a field it's now it's a ring but let's not it's it's no longer a field there's infinitely many of them the question i have is does this tell us anything about finding the parker um square in general or through the true the true yeah and sadly we still don't know so all they can say is you can have arbitrarily large finite fields of a prime order and they will always have a magic square of squares in there they're all non-parker because they work we still don't know if a magic square of squares just without needing to be mod anything exists so still an open question and so um the research will continue i mean the greatest minds in mathematics will continue to turn their attention using things like finite fields and any mathematical tool at their disposal in the soul like the greatest search in maths is are numbers parker or are numbers this discussion continues for a bit longer in a bonus video on our extras channel number file two that's what is the oh my goodness we never parker prize well what are the other prizes out there there's like millennium millennium fields that's a bit rich for more about what matt parker's up to from live shows to books to his own youtube channel stand-up maths check out the various links in the video description could we even communicate with beings from a non-parker reality it boggles the mind

Channel: Numberphile

Views: 94,156

Rating: 4.9655228 out of 5

Keywords: numberphile

Id: FCczHiXPVcA

Length: 17min 25sec (1045 seconds)

Published: Thu Oct 07 2021

Reddit Comments

[u/CrabbyBlueberry]

I was slightly irked by Matt's incomplete explanation of fields, but still a great video.