 Ok so my struggles with windows are continuing, let's hope that they don't do any more crazy stuff. Alright so those things are quite non-trivial in this stack you know, is that it? It's not recording. It's not recording. What part is it recording? Are you sure? It's recording man. Are you sure? The audio is also moving. See there is this better thing in green that's going off. Can you see this? That means it's recording. Ok. Alright so is it ok? It's ok no? Ok. So this is quite a non-trivial this stack. I want you to think about it for a while and it's something rather interesting but of course the multiplication is an F. Remember that it's an F and this has to be true. In fact one can extend this a little bit and give it slightly more interesting because the only thing I'm saying here is non-zero. How do you include zero? Ok so what you can do is if you multiply by x on both sides you just include zero. So basically x power n minus x would factor as product of, let's say alpha in F is minus alpha. Ok so all the elements of the field. Ok so it's quite a non-trivial design and so this is interesting. So I want you to keep this in mind. Ok so we're going to exploit this result quite extensively to understand our field much much better and in fact understand polynomials over finite fields much much better. Once we get to the bottom of it we'll see the polynomials over finite fields are very in a very simple objects. You can quickly find out where the roots will be you know you can easily enumerate everything about them. That's the purpose of this property. Ok the fact that these things happen. So before that I want to give a quick result. If you have F doing an arbitrary field and if you look at some polynomial for T of x on Fx ok and if you factor it into reducibles ok over F or any other field for that matter it could be a larger field that contains that force. Essentially that factorization is unique ok factorization into reducibles is essentially unique. I have to say essentially because you can always pull out scalars in the factorization and then make it look different that up to the scale I'm not going to be playing outside it is essentially unique. Ok so in Fx there is unique factorization. Fx factors into reducibles in a unique way over I'll say some Gx where G is an extension of F. So this is like the factoring of numbers ok so if you have any integer it factors into primes in a unique way. There's no two ways in which it can happen. Same way for polynomials also it's true ok. So this fact along with the fact that the polynomial X power T power m minus x factors in a certain way is very useful and powerful to visualize ok so it gives you a lot of handle on how the polynomials over finite fields work. Ok polynomials over finite fields are quite important for classical codes should have a good feel for what they are and this fact is very important. We'll see I mean as I develop it further you'll see that it becomes more clear but at least initially this is the power ok. So because you have unique factorization over fields over polynomials with coefficients from fields and this polynomial X power T power m minus x factors in this specific way we have a lot of powerful results for polynomials. In fact for instance a degree m polynomial over a finite field you will know exactly where to find all its roots. Given the degree of a polynomial you will know that there will be one field ok some extension of the field of coefficients possibly in which all the roots of the polynomial are ok. So it's an interesting kind of reason it's not as interesting as the results of complexes right for complex number you know but for every polynomial the complex coefficients has one root in the complexes ok so that's a very powerful statement you cannot make a statement like that there's no one field which has all the roots but you can say precisely which extension field will have the roots ok. So that gives you a very nice handle on this. S of X also G is some extension so S is a trivial ok see it might be I'll come to it so that there will be a polynomial which is probably say factors into two polynomials irreducibility. You go to a larger field maybe the number of irreducibility will increase we will have more roots ok. So the factorization will change depending on which field you go to but in any given field it's unique you cannot have two different particles that's the idea alright. So fields are not the only objects over which this happens in fact algebra will see that there are a few other objects on which factorization is unique and it's an interesting field of study so people looked at that also. Alright so let's proceed and so to understand our structure further ok so we have an element we know it's you know that there is a primitive element there are also other elements which may not be primitive ok but there are all these elements and the multiplicative order is an important idea that you should keep in mind ok so the order of the multiplicative order of any element or like I said an order of an element will divide p bar m minus 1 so that much we know and that's a crucial idea ok. So the next thing is to understand polynomials because you remember the reason why I am going again to polynomials is we started the specific construction with the polynomial ok so this is the general field and then I gave you a specific construction what is the specific construction ok. The specific construction started with the polynomial ix which was degree m and it will do so the level is at p right so this is what we started with and then we said this object is actually f p p bar m ok so we said this way which is what a 0 plus a 1 x a m minus 1 x bar m minus 1 and the a 0 coming from is f p ok what are the addition and multiplication addition is simply modulo p polynomial addition right and what is multiplication multiplication is modulo pi x right and maybe I used alpha did I use pi alpha I might have used alpha it doesn't matter what I used right so if you do modulo pi x then I actually showed that this is a field with p bar m elements ok so first of all to make sure that this construction is even feasible I should have a pi of x ok and then depending on different maybe I will get different fields all these questions are still open right so we don't know how many such fields are there maybe there are some other construction how do we know ok. So we will essentially show that all those things are not possible you will show that there is a polynomial function f p bar ok so it's a powerful result and to get that we should know something about polynomials ok so that's the next thing we are going to look at and the critical idea connecting polynomials to fields is this notion of a minimal polynomial as an element ok so suppose I have so we are going to talk about minimal polynomials ok so the crucial idea ok suppose I have beta belonging to the finite field ok let's say that f has so the notion here is a bit more general that I am going to define it in a specific way so let's say f is p bar m ok so we will look at this and we will design the minimal polynomial of beta over z p is the least re-paranormal and so it should be in for p x but beta over root ok so there is a lot of things in the definition that you have to digest so let's walk through this a bit slowly so first of all the minimal polynomial is for an element d tau which belongs to what which belongs to the field f ok but then I am saying the minimal polynomial is for beta over z p ok which means the polynomial I am looking for should have coefficients that are only in z p cannot have coefficients outside of z p which could be in s ok so if I don't put that restriction if I say minimal polynomial of beta over s the answer is a bit trivial what will be the answer what is the least degree polynomial with coefficients from s which has beta as a root x minus beta ok I know already ok so it's not a big deal I don't have to worry about it and that's a valid definition actually but then I am not asking for I am not allowing the coefficients to come from I feel that I am saying the coefficients has to be only from z p ok then I have to worry will there be any such polynomial I am asking for the least degree polynomial first of all is that any polynomial for which coefficients are from z p and beta as a root do you know of any polynomial x power beta over m minus x I know for that beta as a root ok so this definition makes sense because because beta is a root of what x power p power m minus x which actually belongs to z p x the coefficients are from z plus 1 and minus 1 they immediately belong to z p there is no problem ok so that is that is that is the reason why this makes sense but then beta may be a root of a smaller degree polynomial it may not be at all degree p power m, p power m may be very large maybe there is a degree which is smaller than that ok so the minimum problem is the smallest such thing but then I have to make it a bit more specific here ok I can't say they are minimal polynomial till I make it unique the only problem there is I might be able to multiply by scale as and get different polynomial so what usually is done is people have this word monic what is monic the largest degree term is 1 the largest degree the largest coefficient of the largest power term is 1 ok so you choose one such kind ok you cannot show you can not you can not show that the minimal polynomial is unique ok I can say they are minimal polynomial there are two such polynomial what can I do they are both monic both have betas of root and I can subtract one from the other I will get a strictly lower degree polynomial for which also beta will be a root and that will highlight the minimality of the degree ok so I can show that it is minimal polynomial so it is ok to define that ok so all these things you have to pass when you are going to write a statement like this ok in engineering you might be used to have to accept things like statement if you want to be a little bit more thoughtful you have to make sure that all these things make sense and all of them do make sense there is no problem ok so if you want an example the BSP this is just a new passing example as I can give you ok so usually if you look at root stuff for your filters right so you look at zeros and poles for your filter and if you want your filter to have real coefficients that you still want a complex root what do you do ok so you construct a real polynomial for which that is a root that what is the penalty you have to pay they also take the complex conjugate root ok so if I have for instance i the element i which belongs to the complexes what will be its minimal polynomial over complexes bx minus i what will be its minimal polynomial over the reals except this one ok it will have degree 2 side ok but interestingly in complex the minimal polynomial will always have either degree 1 or degree 2 if it is real itself then the degree will be 1 if it is not real the degree would be 2 you know the degree 2 always enough so here we will see some interesting examples ok so the question is we are going to now look at minimal polynomials and without explicitly finding the minimal polynomials we will have some properties for it ok and then we will see how easy it is to find it it is not very hard ok so first of all notation so I will usually use m of beta for minimal polynomial so there are lots of things that I consumed they are not said over what field it is ok so usually it will be clear and so I am assuming I will just put m of beta otherwise you will have to say mp of beta do a mild generalization of these definitions there it will become a little bit more complicated but for now we will just keep it as m of beta m of beta is the minimal polynomial of beta assuming that all the involved fields are here ok so there are quite a few interesting properties for this the first property we will do it in any particular particular field ok so it is fine ok so first property is f of x belongs to belongs to some of the fields and f of beta is 0 then m of x m of beta that is a bad notation I am sorry I want it to be m beta of x ok sorry it is a little bit confusing some x in the polynomial ok so m beta of x then m beta of x will be varied f of x so that is the first result we will show so if you have a minimal polynomial so and if you see that there is some other polynomial f of x in zpx the coefficients of f of x are in zp and if somebody tells you that beta is actually a root f of x then what has to happen this entire minimal polynomial should be a factor of f of x ok so this is an interesting result to show any ideas on doing this yeah I would use reminder so I would use division with reminder so you take f of x and divide that m beta of x you will get a quotient and a reminder reminder is strictly less than the degree of m beta of x now we put that sequence beta so you see that the reminder has beta as a root so that you violate if f of x is non-zero then that you violate the minimality of the degree that you had for the minimal polynomial so you just use the definition and prove this to somebody so division is what we use and you prove I am not writing down all the steps but this is important ok I will do a few properties and then I will give you an example so I know examples have not come so far so few properties are good and then we can do the next property is degree of m beta of x is less than or equal to m so remember I will do m m came from here size of f is p power m if m was like the power of p so you might wonder if there is an example I told you that maybe the degree of the minimal polynomial was very very large p power m or something and so that's only at most m so the proof of this is a little bit more in world let me see any ideas how to prove this well that's a bit hard I don't want to use diafx now see I don't know if my field now has constructed from the diafx don't think of the specific construction I don't want to use those properties this is some abstract finite field that somebody told us there and all the definitions I made are with respect to that field so this is not necessarily for those specific constructions so here the trick is to use the vector space figure so what we do is you look at these these elements 1 beta beta squared 1 to beta power m minus 1 and then beta power m ok these are all elements of work there are elements of s and s is a undimensional vector space over zp how many elements do I have here m plus 1 so what should happen definitely they should be linearly dependent ok so these are m plus 1 elements of a finite field of a vector space which is only m dimensional over zp which means it will be linearly dependent over zp ok so these are linearly dependent over zp ok and that's enough to prove my claim what does that mean that means that exists what does this mean linear dependence that exists a0 a1 am in zp such that what such that a0 times 1 which I will call as a0 plus a1 beta plus a2 beta squared plus 1 to am beta power m equals what 0 which means I have explicitly showed a degree m polynomial for which beta is a well I am not explicitly showing that just showing the existence of a degree m polynomial for which beta is a so I can conclude that the degree of minimal polynomial for any element in my finite field s has to have degree less than or equal to m cannot be greater than m cannot show this is equal to m though only less than or equal to m it could be a lower degree polynomial and that's QED alright yeah so once again don't get confused by the specific construction I told you that that's a unique kind of construction we can prove many results for that abstract finite field that we had and we want to show as many results as possible okay so this is the second property and let's see the less one more nice thing I can show but maybe I should prove something else yes, is there any question okay so the next property is quite interesting okay so so yeah so let me show this okay so suppose you look at m beta of x it turns out m beta of beta power p is also equal to 0 okay so that's the next property okay so you take beta and raise it to the power p p is remember my characteristic and then you substitute beta power p in the minimal polynomial for beta so that beta power p is also root for the minimal polynomial okay so how do you prove this proof goes like this it's not very hard okay so the first thing to show so you need a kind of a lemma so one of the things you should know is the following okay so if you have any polynomial let's say is 0 plus okay so let me start with the simplest okay x plus y power p is actually equal to x power p plus y power p for x y in okay so if you have two values x and y that take values in the finite field s which has characteristic p I take x plus y so this is like I have to prove this so if I take x plus y and raise it to the power p I will get p plus y power p all the other terms will go off to 0 the reason is if you take any other term that some term in the middle it will have this form p choose i so what is p choose i p times p minus 1 sum to p minus i plus 1 divided by quantity 2 into sum to value okay what about this p you know it's a prime number so it will never get cancelled by any of these but of course I know p choose i is also an integer so somehow all these cancellations will happen with the other terms that this p will always remain so this is actually sum a times p a times p so it's a multiple of p which is equal to 0 in x so all the other coefficients that you have for the other powers they will all go to 0 in x it's a bit difficult to think about it but that's true because also because p is prime if p is not prime then this won't be true if p is prime it's a characteristic of the field so any p choose i is actually a multiple of p and goes to 0 in s so x plus y raise to the power p is equal to x power p plus y power p so now I can generalize this so if I have say x plus y plus z raise to the power p what will I get x power p plus y power p plus y power p that's very easy to do so you split it as 2 terms raise it and then raise it further so what happens is if I take any problem on L 0 plus a1 plus a1x plus a2 x squared plus a12 say some a m that's power m and then raise it to the power p what will happen in this way I can do it as long as I know that I am only going to put values for x and a which are in fields of characteristic p I can individually raise the power here so I will get this here so I get a0 power p plus a1 power p x power p plus a2 power p x power p now in addition if I tell you that ai belongs to zp okay whether you just simplify further yeah so in zp what do I know ai power p over ai power p will be equal to ai in zp right so if we element raise to the power p is equal to itself so that's true in zp and any field for that matter right so remember we had this equation x power p power m is equal to x in x right so in any field if I raise it to the size of the field you will get the same element so this actually becomes equal to okay so if you call this polynomial as dx this is actually af afx for p for afx in dx so this is equal to af that's not enough to do my statement I have to keep going further okay so I am going to use this now I can repeatedly use this so if I do it once again what will happen if I do afx power p squared what will happen afx power p squared only in x you will have the power and this is true in one way so afx power p power ai equals afx to the power p power r okay okay so okay so looks like I shouldn't have jumped to this property directly but anyway so that's a good property to show okay so now so is this enough to show the result that I want that's fine that's fine is enough why okay okay so maybe this is enough okay so maybe so now if you use instead of afx if you put afx you set afx equals mbeta afx in that property what do you get mbeta afx raised to the power p equals mbeta afx power p right now you put x equals beta what happens on the left hand side you will get 0 on the right hand side you will get what you want okay so put x equals beta okay I mean beta af beta power p to be equal to so now you can repeatedly do this do this okay so beta is a rule beta power p is a rule I can in fact generalize this a little bit okay so in fact there is a more general statement okay more generally the statement is true so if you have afx and beta p any polynomial not necessarily minimal or something if alpha in f is a rule plus afx then what else is true alpha power p is also okay so this is where more general statement I used it specifically for the minimal polynomial to get one result but in general it's true because the coefficients are in set p if you have a root in x then that alpha raised to the power p is also a rule so now I can repeat it okay so alpha power p now is a rule which means what alpha power p squared is a rule so so on I can keep on going till that alpha power p for something becomes 1 so I can keep doing that so I don't know where it will become 1 it will become 1 somewhere it becomes 1 if it is a primitive element I didn't say primitive it's not primitive okay so now we can repeatedly do this remember so alpha power p is a root so so on alpha power p squared is also a root and so on okay remember that's true I am sorry need not come also I will come again in the sequence whatever it comes you won't get anything new whatever it comes okay so the idea of this is called what's called conjugates this notion of conjugates that you have seen in general over the complex of the reals now you can generalize so if you have alpha belonging to f then alpha power p alpha power p squared so on are conjugates of over zp so these are called conjugates because for any polynomial for which alpha is a root these also have to be roots so the same thing is true in the complex over reals right if you have a real coefficient equation and if you know that a complex number is a root then what else is a root the conjugate is also a root so same thing here except that now there might be more than one conjugates for each other I think we are coming up to 11.30 so I am going to stop here for today so keep this in mind we are looking right now at minimal polynomials and these will be going to give us important connections to the general construction the specific construction and then we will generalize ok so we will start from here