okay so welcome to this next video in the play Gaston vector spaces in this video what we’re going to talk about is Morpheus basically okay so in the previous video in this playlist on vector spaces we looked at the definition of a field what we’re now going to do is have a look at some more examples of fields now in the video on the definition of a field I gave you free examples of fields okay and those were the rational numbers the real numbers and the complex numbers however those are really really complicated sets and they’re really really complicated to construct what we’re going to see in this video is some examples of some much much simpler fields in fact these are going to be fields with a finite number of elements in okay and we’re going to have a look at the prime fields okay right so firstly what I would like to do then is just remind you of the definition of a field then we will go through the construction of a prime field ok so firstly let me just remind you then of the definition of an abstract field okay so an abstract field which we’ll call capital F here okay it starts its life off as a set of symbols okay so you firstly start off by creating a great big set of symbols which can either be finite or infinite okay so that’s the fundamental starting point for building any number system that you have to firstly create your set of symbols okay right then what we’re going to do to this set of symbols is we’re going to define on it to composition laws okay now the first of these composition laws is going to be called addition okay and the symbol for this is going to be plus and you’re going to be able to compose any two elements of the field together under this composition law of addition so we’ll have a great big compositional table here for addition okay so I’ll color this in so I’ll have addition represented in Korean here okay like so and you’ll give all of the elements of the fiord a row in this composition table so you put all of the elements of your set here down here you’ll give every single one of row dedicated just to it and you’ll give all of the elements of the field a column dedicated to it as well and then what you can do is go through and fill in all of the entries in this great big table and define thus what any element of the fields compose with any other elements of the field is equal to under this composition of addition okay and we know that the axioms that this addition composition table needs to obey is the axioms of an abelian group okay so addition must obey the axioms of an abelian group okay so what that means is that if you just consider this set of symbols which is the set of symbols of the field here with this single composition law of addition on it then the number system that you have created will be classified as an abelian group okay now to turn this abelian group them into a field what we have to do is put another composition law on top of this one okay so we have to define a second composition law and the second composition law is called multiplication again we’ll have multiplication cut it in in orange so again what you need to do is create a composition table for this so I’ve drawn my composition table here and the symbol for multiplication of two elements is a times of course and what you’ll do once again is you’ll fill in all of the elements of the field over here so you’ll give every single element of the field a row in this multiplication composition table here okay and you’ll also give every single elements of the field a column in this multiplication composition table so all of the elements of the field will be up here at some point and then what you can do is go through and define what and the element of the field multiplied by any other elements of the field is equal to four then all of the entries in the great big multiplication composition table and if this multiplication composition table obeys certain axioms then we will have created a field Okin I’m just going to remind you of the six axioms then that this notification composition table must obey in order for your number system that you’ve created here to actually be considered a field so that starts over here so axiom number one then axiom number one is closure okay it says and I’m sorry about the power cut there it should be fine because my time working on a laptop so it’s for the battery but the lights hopefully will come on at

some point okay we’ve got natural lights that should be fine okay right so axiom number one either oh dear they’re going on and off right so axiom number one men of multiplication says that if you take any two elements of the field okay little X and little Y and you multiply them together so I’ll just put a for all little X and little Y are elements of the field so for all the little X and there’s a wire elements of the field if you multiply x and y together okay and the way that we usually usually denote two elements multiplied together is we just write them next to each other rather than actually sticking a multiplication symbol in between okay so x times y here that answer must be an element of the field so this is the closure axiom okay and what this means is that all of the entries in this multiplication composition table here must be within the field capital F so you don’t have any answers in here that are outside of the field you don’t just suddenly make up a new symbol and stick it in there as one of the answers okay that’s a lot of a house all that the answers must be back within the field so that’s a fairly intuitive axiom and as they say the name for that of course is that multiplication is closed out there we go closure okay right so there’s axiom number one okay now we’ll go on to axiom number two axiom number two is associativity now this is a complicated property but we’ve studied it to death in group theory so we’re used to it by now okay so associativity concerns multiplication of three elements of the field so for all little X little Y and little sad that you can pick from the field it must be the case that if we consider the multiplication of three separate things X multiplied by Y multiplied by said there must be one and only one answer to what that is now when we consider multiplication of three things obviously we can’t just do it with this multiplication table because the multiplication table really tells us how to multiply two things together at once so if we want to multiply three things together the way we have to do it is by putting some brackets in here to split it into multiplication of two things at any one time okay but the problem is that there are two ways of putting in brackets you can either put the brackets around x times y here which means firstly multiply x and y together which is something you can do using the multiplication table take the answer which by axiom number one is another element of the field okay and then multiply that answer by Zed and that’s another multiplication of just two things okay or the other way that you can put the brackets is you can put the brackets around Y times that so this would mean firstly multiply y&z together take the answer and then let take X and multiply by the answer okay and the associative ‘ti is basically the property it’s the law that says that these two things must be equal to one another and that’s not trivial at all okay so you’ve just made up the answers in this multiplication table even if you were closed okay so all the answers you put in worth and a field the likelihood that it would have a associated very low unless you actually knew what you were doing okay so this is a very complicated property okay but really the gist of it is saying that if you’re multiplying three things together there is one and only one answer to that it does not matter where you put the brackets to turn it into something that you can actually do with your multiplication table which just involves the multiplication of two things okay write that next up let’s do axon number three then so multiplication must have a associativity next up axiom number three and wait at the moment just going through these axioms in exactly the same order as we went freedom in group theory and indeed this multiplication table must obey quite a few of the applications of group theory it is almost a group composition table like addition there is going to be warm twist in axiom number four as we’ll see okay right so axiom number three so maximum of three says that the Mystics is the multiplicative identity so there must exist an element which we’ll call one okay which is an element of your field such that one can multiply by any other element of the field both way rounds to give that other element of the field back again so one times X and this is one of the few places where I will put the multiplication symbol in between the two elements okay 1 times X must equal X and also the other way round x times 1 must equal X and that must be true for absolutely all X that you can possibly think up from your field

capital S okay so that’s the condition that there must be an identity element of the field for multiplication okay now there will also be an identity element for our Edition composition law and of course we call the identity element for addition composition or zero so two of the elements that you will have here are 0 & 1 0 is the additive identity the identity in this composition table so there’d be the additional compositional and one is the multiplicative identity the identity in multiplication composition law okay right so that’s axiom number three now acts in number four and this is the one which does actually vary from at the axioms of brut theory ok the axioms that this addition composition or must obey in this concerns multiplicative inverses okay so axiom number four says that for all elements of the field so for all the 2x that is any element of the field except it can’t be one element of the field I mean only elements of the field I’m not going to insist that this is true for is the additive identity seen a row so you take any other elements of the field bar zero and I’m now going to insist that there is a multiplicative inverse for it okay so there must exist some other element which we’ll call X to the power of negative one or you can call it one over X if you like which is another element of the field and this is going to be the multiplicative inverse of X such that x times X inverse is equal to the multiplicative identity 1 and X inverse times X is equal to the multiplicative identity 1 so you can multiply these two elements either way around and you’ll get the multiplicative identity back against that a must exist multiplicative inverses for all elements except 0 ok and we saw in the previous video when we went through the definition of the fuels are more slowly that you can’t have a multiplicative inverse for 0 if you’re going to insist on distributivity because the instant you insist on distributivity which we haven’t got to yet it will be acting number 6 instant you insist on distributivity the zero must multiply by everything every element of the field to give zero so there can be no element which zero would multiply two with to give 1 basically okay so you quite simply cannot have a multiplicative inverse for the additive identity if you want distributed that is hold and distributivity is more important than having a mult it’s inverse for the additive identity okay so that’s axiom number four then of that this multiplication table must obey axiom number five is a very nice and simple one it’s commutativity okay so it says that for all elements x and y that you pick from the field it must be the case that if we take x times y it’s equal to y times x so that’s basically saying that this modification composition table is symmetric down the diagonal line it’s commutative okay right so underline that one I think in red and then one final axiom which is axiom number six and this is distributivity okay so this says for all elements a little X little Y little Z that you can pick from the field it must be the case that if you consider what is X x the answer to y plus said that that is the same as x times y plus x times says okay so I put brackets around here and here implicitly there are brackets around those two so it’s basically saying that you can either firstly add these two together and then multiply or you can multiply first and then add over so this is the condition that multiplication distributes over addition again this is the one that links to composition tables of addition and multiplication together and makes fields very very interesting and we use this property all the time and field theory okay right so those are the axons then of an abstract field so now what we’re going to begin then is our construction of the prime fields okay but just before we actually do begin our construction of the prime fields I just want to discuss something that we’re going to use in our construction of the prior fields so I told you in the previous video that Irrational’s with position law of addition and multiplication on them where a field the real numbers with the composition laws of addition and multiplication on them where field the complex numbers would be compositions of addition and multiplication on them where a field okay what about the set of all integers

with the composition laws of addition and multiplication on them okay so you do indeed have addition and multiplication defined on the set of all integers just remind you what the set of all integers is it’s the set of all whole numbers that’ll be 0 1 negative 1 2 negative 2 3 negative 3 okay and you’ll go on and on a lot of course ok so here is the set of all integers now under the composition law of addition the integers is an abelian group ok so it certainly has an addition composition law on it which is in the beginning group okay so now let’s think about the multiplication that we define on top of that so we can define a multiplication of integers together and let’s think about whether or not it’s a field and the answer is that’s a lotta field but let’s think about why it’s not a field ok let’s have a look at axiom number one so we must be able to multiply any two elements of the field together to get another element of the field well that one is satisfied by the integers and multiplication because indeed when you do multiply two integers together you get another integer when you multiply two whole numbers together you don’t suddenly end up with something that’s not a whole number okay so indeed all of the answers are back within the set so that one satisfied so the integers under multiplication is closed okay axiom number two associativity is certainly true we use that all the time okay so if socia tivity is going to hold the identity we’ve got one which we’ll multiply by any other integer to give that other integer back again okay and here’s where it doesn’t hold axiom number four that all elements except the additive identity must have a multiplicative inverse okay well we can quite simply look at two okay look at two here does to have a multiplicative inverse in the integers well of course the answer is no there is no integer that you can multiply two by two give one okay of course the answer in the rational numbers will be a half but a half isn’t in here so two does not have a multiplicative inverse and in fact all numbers except one a negative one don’t have multiplicative inverses in the integers so it fails axiom number four big time and that is in fact why the integers with addition and multiplication on them is not a field okay right but let’s just make sure that it obeys these final two axioms five commutativity if we do multiply two integers together you can do it either way round and the answer is the same it does not matter what way round you multiply two integers together the answer is the same so commutativity of course is true in the integers and distributivity we use that all the time as well so this is also true in the integers as well so in fact the integers obeys all of the same axioms as a field except the fact that it doesn’t have this requirement of all elements but the additive identity having a multiplicative inverse that’s where it fails okay so the integers is actually classified as a commutative ring okay so that’s the fancy name for what the integers actually is okay and the commutative ring obeys all the other axioms of field theory except this one here okay but we are going to use this commutative ring of integers in this construction of the prime field so I feel it’s important to just say that this algebraic number system does actually obey a lot of the familiar axioms because we’re going to make use of them in our construction of the prime fields okay so that’s why I’ve just pointed out that apart from not having multiplicative inverses this is very nearly a few basically Oh again doesn’t either Bay a lot of the axioms that we are going to use later on okay right so that’s now construct the prime fields so firstly let’s start off by giving the name of the prime fields okay let’s give them a name before we even constructed them okay so the prime fields are going to be called at this beautiful symbol this strange F here like say where you have two lines it’s kind of like the real numbers where you have those two lines in the R okay so you have this strange F and then you subscript it’s a little P here okay so this is how the prime fields are denoted okay and this little piggy can be any prime number you like okay so the little P can be two it can be free it can be five it can be seven it can be 11 it can be 13 etc you get the idea all the prime numbers the numbers that are only divisible by 1 or themselves okay so you

can let P vary over any one of these that you like and if you use any one of these different primes to make the prime field corresponding to that prime you’ll end up with a different prime field basically so for each prime number there’s going to be a corresponding prime field denoted as strange F subscript that Prime okay so I’m going to construct it for a general prime okay rather than doing it for a specific one so my arguments are going to apply for absolutely all of the prime numbers which is of course the best way to do this right so how are we going to actually construct these things well the reason I’ve just been over the definition of a field again is to give us some insight into how we’re actually going to go about constructing a field ok and the key thing that we start off with is what an abelian group okay so that’s firstly try and build an abelian group and then let’s define multiplication on top of that abelian group and that’s hope that it will end up obeying all of these six axioms that the multiplication the one must obey so we’re going to start off our repeat it by trying to construct an abelian group okay and before we actually go any further let me just say that once we’ve actually constructed this you’ll see that this is actually going to have order equal to the prime P okay so the number of elements that each of the prime fields is going to have will be that Prime that we use to construct it basically so f2 will have two elements f3 will have three elements F 17 will have 17 elements okay right so let’s now begin the construction then so we’re going to start off by trying to construct an abelian group okay and the abelian group that we’re going to use to build these prime fields is going to be a quotient group of the group of integers okay and this is why we spend that little time talking about the integers mod it out or quotient about by the subgroup he’s said okay right so I’m just going to go over this okay I’m gonna remind you how you actually construct a quotient group of one group by a normal subgroup of that group okay so at the moment we are just thinking about group Theory not feel a family or in the commutative ring theory okay we are just thinking about the integers as a group okay so we’re just thinking about this set here of all whole numbers with the addition composition law defined on it forget the multiplication composition law for now okay and under addition the integers is just an abelian group okay now what we’re going to do is we’re going to take the quotient group of that group of integers and tradition by this subgroup which is denoted PZ now let me just reminder what that actually means so this is the subgroup that consists of all integer multiples of this prime P so it’s going to it consists of all things of the form some integer integer little zared here times P so little add is going to vary over any element of the integers here okay so that’s the definition of this subgroup P set as it’s all integer multiples of P if I write that out a little bit more explicitly there still set that contains well let’s go through the different integers here so start off with 0 so we get 0 times P which will just be 0 okay and of course we are using the multiplication ball here to construct this subgroup okay so again I’m relying on you understanding from classical algebra how to multiply two integers together okay so one a times P you will get P negative 1 times P we’ll get negative P 2 times P will get 2p and you can go on then we’ll have negative 2p 3p negative 3p 4p negative for the etc etc and this is in fact a subgroup of the group of integers and tradition so it’s closed whenever you add two multiples appear together you get another multiple P and there will be a associativity because the larger group obeys associativity it has the identity element 0 in it so it will have that and it also has additive inverses in it so for any element you also have the negative of that element which will be the additive inverse okay so indeed this is a subgroup of the group of integers under addition okay right now because we’re working in an abelian group because the group of integers under addition is indeed an abelian group it’s commutative okay all subgroups are normal subgroups because remember in a beating groups conjugation has absolutely no effect of whatsoever okay so that means that whenever you conjugates elements or

subgroup by elements of the group you will always end up with that same element back again which is certainly an element of the subgroup so all subgroups are stable under conjugation by the larger group basically okay so that means that this subgroup P Zed is in fact a normal subgroup of the group of integers and that means that when we construct the left and right coset partitions of the group of integers and tradition under this under this sub group here we will get the same and so basically the right and left cosets are exactly the same so there’s only one way of partitioning this group up into the cosets of Lisette okay so the right coasts and the left coasts are the same we will just use the notation for left cosets but we could have used the notation for right cosets okay so remember when we want to construct the quotient group of Z by this subgroup P said what you now do is you you divide up the larger group you partition of the larger group into Co sets of the normal subgroup here and then the elements of this new quotient group are going to actually be those Co sets and then you can define a addition on this set of Co sets and that’s going to be well defined as we’ll see okay right so let me draw a picture for what we’re about to do here so I’m going to draw a number line which is the intuitive way of viewing the integers okay so I’m firstly just going to mark on the elements of this subgroup PZEV so here we have 0 here we have P here we have negative these I’m just marking on the multiples of P okay then we’ll have 2 P over here and I’m trying to do it as neatly as possible we’ll have negative 2 P over here and of course we could put 3p and negative 3p on and we’ll go on forming this beautiful even lattice in this way okay like so so here is our actual subgroup he’s dead okay and now what we want to do is want to partition up the entire number line the entire set of integers into the Co sense of peace Adam will use the notation of left cosets although as I keep stressing we could use right Co sets they’re exactly the same okay so what we’re going to do then is we’re going to take all the elements in this subgroup P said and we’re going to now add on one so we’re going to form the left coset of P said under the element 1 here so we’re going to now add on 1 to all of these who will then get one here will get P plus one and this is gonna get a bit squashed and then over here we’ll get two P plus one and over here we’ll get negative P plus one will get negative two P plus one over here and etc okay so this is going to be the set of all elements just to the right of the elements in this subgroup please add here okay so this is the left coset of the subgroup P said under the element pot okay and then you can reach him you are you can then do the left coset of peas head under two again what will that do that will take you on to the next ones along so let me just mark these on i won’t try and put the annotations this time because otherwise they’ll get too crowded okay i’ll color these ones in in orange i think so here are the elements of this left coset of peas add under to this time all of the elements of peas add moved to the right by two this time and you’ll continue on so you’ll have three + pz 4 + if you said right up to what will be the final co set of we will need to cover the entire integers well that will be the one where we’ve shifted everything along by P minus one so we’ll go all the way along to P minus 1 plus P Zed and when we shifted all of these multiples of peas head along by P minus 1 where will they now be on this number line well they’ll be just now to the left of the one ahead of them so this one will be moved up to here okay so this is 0 plus P minus 1 basically this one here which was P will be moved up to here so again now this is to P minus 1 now ok so this is to P minus 1 and then this one will be moved up right to the left of 3 P up here I’ll color these in as they go along okay so here they are this one here negative P will be moved just to negative form here like so and negative 2 P will be moved up to here so negative P minus 1 here okay right so that then will be our final left coset of PZ and this time it’s under the other than P minus one so

here then are all of the left cosets that we can partition up our group of integers into using this subgroup Peas dad okay so what we’re now going to do them to construct our quotient group of said by peas ed here is we’re going to stick all of these Co sets into a set and that’s going to be the set that’s going to form a new group a cigar new what we’re going to see is going to be an abelian group okay right so we need then a name for all of these left cosets of the subgroup peas dead okay so we’re going to use our old notation from group theory okay and a common notation for denoting these Co sets which are also called equivalence classes is to just take some representative from each one of the co sets and put a bar over it to denote the co set okay and we’re going to pick clever or simple choices okay so if we wanted to do it for peas add say we wouldn’t pick a hundred and one times Pig with me that would be a representative but it wouldn’t be a good choice of a representative the most obvious choice to pick for the representative of peas head is this element 0 here okay so we’ll call this 0 bar the co set of the subgroup peas add that contains 0 bar and the subgroup itself is a co set of itself it’s the co set under any element that’s actually in it okay so we’ll call this 0 bar then for this one 1 plus P Zed again we’ll pick a sensible choice and the most sensible choice to pick would be one here okay so we’ll call that the coaster that contains one so we’ll call that one bar okay and then we’ll go on in a similar fashion so for the co set 2 plus P Zed we’ll call that to bar the co set that contains 2 and we’ll go on and on all the way down until this one will be cool the co set that contains P minus one so it’ll be called P minus 1 bar okay so that’s how we’re going to notate all of these Co sets of the subgroup P’s there so we’re now going to stick in all of these symbols for the co sets of this subgroup peas add into this new set that I’m constructing here so how does ero bar there we’ll have 1 bar 2 bar out all the way up to P minus 1 bar okay so how many elements have we overall got in there ok what’s going to be P elements because from one bar to P minus 1 bar we have P minus 1 then when we add on 0 bar here we have P overall ok so this is looking hopeful considering the fact that the prime fields I’ve already said are going to have order the prime P okay so we have an in fact but our set of symbols now okay this is the set of symbols that is going to form our prime field ok so say hello to the symbols right so now what we want to do is put the law of addition on here okay and the law of addition comes with this operation of taking the quotient group of the group of integers by this subgroup PZ okay so there’s always a composition board that you can put on a quotient group basically okay and in this case it’s going to be the addition 1 ok right so I think I’ll have a break here and in the next video we’ll firstly start off by looking at the way that we can define addition on here ok we’ll remind ourselves of why that law will be consistent and we’ll go through why obey the axioms of a meeting group and then what we’ll do is move on to how we can define multiplication on top of this abelian group of Z by P set