 So, now we will look at a proof of this. If I put this through to you, what is your best guess for a potential generator of any ideal in the commutative ring of integers sorry in the commutative ring of polynomials? What is your best guess? Suppose I give you a subset listen this subset is not any arbitrary subset right, it is already an ideal that has been told. What is that property of an ideal that is so interesting and how can we harness it to sort of narrow it down to just one polynomial and I am saying that look if I told you this one polynomial, you can go ahead and generate this entire ideal. Think about the analogy with numbers. Do you think even numbers in the ring of integers yeah? Do you think even numbers form an ideal? Even numbers do form an ideal? What is the generating element of the set of even numbers? 2. What is common? What is what is so special about 2? It is a common factor of course. So, then if I treat in I mean if I treat polynomials in the same manner as I treat integers, what would be your best guess for the generator of any ideal? But one is not interesting right? We would want no of course not. So, we would want some non-trivial polynomial. When you are talking about one, it is a polynomial of degree 0. There is no point to it really. So, when I am searching for some generating element excuse me in the ring of in the commutative ring of polynomials, I want at least a polynomial of degree 1 or more. So, why do you think x is going to be? So, let us say I give you x plus 2, what do you do? Can you generate? So, I mean what would be your best guess for any ideal? Yeah, what is your ideal first? You have to you must ask me the question right? Then what is the ideal? Unless I tell you what the ideal is how can you just say x or x squared or x cubed or so on and so forth right, but I am not specifying any ideal. I am just saying the ideal itself has some structure and that structure itself helps me in narrowing my search down to the single generating element. I should not say the as your friend rightly asked whether it is unique a single generating element. So, again the answer lies in the analogy with integers 2 right. So, I am saying look at the polynomial of degree greater than or equal to 1 of course. So, we do not look at constants with smallest degree of course, when I say smallest I have to specify positive because otherwise it would not be integers right that is implicit. If you take negative degrees it is no longer a polynomial is it not? What can you is it is a good guess would not you say so? Why? So, I am claiming this is precisely the or a generator for the ideal ok. I should also specify here in the ideal of course, I am not searching for the smallest degree polynomial in the entire ring right. So, I give you an ideal and you figure out the smallest degree polynomial of degree greater than or equal to 1 at least not the constants yeah and that is going to be precisely the generator for the ideal that is my claim. Any questions? So, that is going to be my claim how do I prove this? Yeah that is boring that is then the just the ideal of you know just the integers why should I even why should I even bestow upon it this quality of being you know polynomials let us just treat them like integers forget about polynomials then it is only when you have non-trivial degrees 0 degree is I mean you just call it a polynomial for the sake of it it does not make any sense right it is just you can treat them like integers. So, it is only when the degrees are greater than or equal to 1 that they become polynomials rightly right. So, the claim is that this is true. So, what do I need to prove essentially in order to show that this is true? Let us contradict this ok. So, suppose f x sitting inside A is the smallest degree of course greater than 0 polynomial in A. Say that there exists g g of x belonging to A such that g does not belong to the ideal generated by f, but g belongs to A right then I would have violated the claim you agree that there I have found you some fellow inside the ideal which cannot be generated by f right oh yeah does not belong to thank you. So, I have found you a g which belongs to the ideal, but it cannot be generated by f when does something fail to belong to the ideal generated by f what can you say about the degrees of g and f first by my very choice f is of the smallest degree. So, the degree of g is at least as much as that of f if not greater. So, clearly degree of g is greater than or equal to degree of f right therefore, I can go ahead and divide g by f and what should happen as a result of the division I should have a nonzero remainder if I did have a zero remainder then of course, g is generated by f yeah because there is some quotient only and there is no remainder. So, if this has to be true if this has to be true by virtue of this I might as well conclude that g of x is equal to some quotient of x I am not writing where these are these are all polynomials you understand yeah some quotient of x times f x plus r x with r x not equal to 0 right, but I am saying I already have a contradiction then why let us use a different color what have I claimed this object where does this come from the ideal play. So, let me write r x is equal to g of x minus q of x times f of x where does f come from a. So, what can I say about q x times f x this object belongs to a what can I say about g x it comes to a. So, this entire object belongs to a this object belongs to a what can I say about the sum belongs to a. So, therefore, r x belongs to a what do I know about the degree, but what do I know about f it is the smallest degree polynomial sitting in the ideal and now I have another polynomial r x which is of smaller degree by my claim and it is still sitting inside a right by my very claim I have chosen f x because of its special property that it is the smallest degree polynomial sitting inside this and now I come up with this conclusion that this fellow r also has to be the smallest fellow sitting inside I mean it is not the smallest it is actually of a degree smaller than f because otherwise this division would not have made sense this is a divisor this is a quotient and you stop at a remainder only when it is smaller than the divisor. So, in this case smaller means a degree just like in integers it is the magnitude of the number when you are replacing integers with polynomials you have to in your mind replace it with a degree. So, you stop a division when the degree of the remainder becomes smaller than the degree of the divisor. So, degree of r is less than degree of f but that is a contradiction. So, therefore, you cannot have the situation where what happens that you fail to find a generator by narrowing down your search to the smallest degree polynomial in an ideal. So, anytime I tell you that look here is an ideal can you pick out an entry in this ideal an element in this ideal which generates this entire ideal all that you have to go for is to search for the smallest degree polynomial in that in that set and you are done right. So, that is why whenever you are living inside this commutative ring of polynomials anytime I tell you this is an ideal of the ring of polynomials you know that such an ideal is a principal ideal that is it can be generated by a single element right ok. Why is this important to us now comes the question of uniqueness. So, of course, if a polynomial generates an ideal what do you think if you multiply the polynomial by a scalar in the field will that not also be a generator. So, to answer your question that you asked about whether it is a unique generator well of course, it is not unique if for example, x plus 1 is a generator of an ideal precisely the ideal of all those polynomials whose root is at minus 1 or which vanishes at minus 1 then of course, 2x plus 2 or 7x plus 7 is also an idea is also a generating element of that same ideal no different you will end up with the same ideal the same set of polynomials. So, it is not unique, but you have to if you have to narrow it down to something unique then what you have to do is to make it monic precisely. So, every ideal in fx has a unique monic generator this is very straightforward again we will do a proof short one suppose not right. So, therefore, you have some ideal a which is generated by g1 and which is also generated by g2 with g1 not equal to g2 and g1 g2 belonging to fx are monic, but what do we know about that particular generating element you see it is the smallest degree polynomial it is the smallest degree polynomial, but if both g1 and g2 are generating this ideal then both g1 and g2. So, clearly g1 and minus g2 because minus g2 is just a scaling of g2 by minus 1 both of them belong to a implies g1 minus g2 also belong to a, but what do you know about g1 minus g2 if both are monic what do you know about the degree of g1 minus g2. So, degree of g1 minus g2 is strictly less than degree of g1 or degree of g2 then what does it become it violates the condition no we have already seen that the generator is precisely the smallest degree polynomial sitting inside the ideal now I find you found you another polynomial of degree strictly less than the degree of the so called generator then that will lay its claim as the unique generator or at least of a generator, but that violates the condition right. So, this is a contradiction so this is a contradiction. So, the moment I restrict it to be monic in addition to saying that it is the smallest degree polynomial sitting inside the subset which is the ideal then there is a unique monic generator for every ideal of a ring of polynomials right and this is beautiful because now we are going to look at some more interesting some more interesting kind of ideals okay. We are going to go back to matrices or operators if you like and in relation to those matrices or operators we will see some interesting ideals. So, this part is clear so far so it is a contradiction because g1 and g2 are both inside the ideal and both are monic. So, you have g1 let us say is x to the n plus a1 x to the n minus 1 plus dot dot dot till a n and g2 so this is g1 and g2 is x to the n plus b1 x to the n minus 1 plus dot dot dot till bn both of these belong to the ideal. So, then g1 minus g2 is going to be what a1 minus b1 times x to the n minus 1 plus dot dot dot till a n minus bn what is the degree of this polynomial strictly less than n yeah but this fellow must belong to the ideal as well. So, these fellows belong to the ideal therefore this fellow must also belong to the ideal by the definition of an ideal but this fellow has a degree smaller than the so called generating element either you like whether you like g1 or you like g2 you have claimed that at least one of them maybe half the class claims g1 is the is the unique monic generator half the class claims g2 is the unique monic generator and then they fight and then we say okay let us resolve this let us say g1 minus g2 right but then g1 minus g2 by the definition of an ideal must belong to the ideal and that resultant polynomial turns out to be of degree less smaller than either of those right. So, that cannot be right unless of course this is just 0 that is the only possibility which is when term by term they must vanish because this must be true for every x so term by term they must vanish yeah clear. So, that is how we have a contradiction. So, now we are going to go back a bit to looking at operators or matrices and describe certain special ideals. So, forget about the use of this next term that I am going to use because we have also used the same term to describe something completely different okay. The term is annihilator I know we have used that in connection with dual spaces this has got nothing whatsoever to do with those objects okay. So, now what we are going to talk about is the annihilating ideal okay the annihilating ideal of A where A is an operator. So, suppose A is an operator from V to itself then here is how we define the annihilating ideal of okay let me just put it in because the subscript will come in handy. So, the annihilating ideal of A I am upfront already claiming it is an ideal although it will require me to show that it is indeed an ideal. So, how do we define this now you might be confused oh this is A how is this related to polynomials and that is where this definition comes in it is a collection of all those polynomials from the ring of polynomials such that when you pass the matrix or the operator as the argument of this polynomial instead of just x okay it gives you the 0 operator this is a 0 operator remember this is not a 0 vector it is a 0 operator matrix of operator yeah. So, this is it now you see this is an object we were sitting inside this. So, you should have no doubt about the fact that this is at least belonging to the right sort of structure. So, it can be you know what do you call it it can be an ideal right. So, this is the annihilating ideal of A collection of all those polynomials in simple words it is a collection of all those polynomials for which if we pass the argument of the polynomial as this matrix or this operator when I say an operator to the power n it means it is a composition n times I am hitting it successively n times with the operator just like in matrix in case of matrix is just a product in case of an operator it is just n successive actions of the operator on a vector all right. So, this is a 0 operator question is is it even meaningful to seek such a thing how do we even know at least in the finite dimensional case that is when the v is finite dimensional I mean we might be seeking like you know let us seek a cuboidal elephant elephant because an actual elephant's topology is very difficult we cannot say anything about an elephant. So, let us assume that the elephant is a cuboid and then prove all our theorems that is a very popular joke among physicists by the way because the actual thing is too difficult. So, we make all our idealizations and build everything on it and then low and behold the moment you step out of a classroom the classroom is the only cuboidal object that you have ever seen and you see that no the elephant is far from being a cuboid right. So, we should make sure that we are not seeking such cuboidal elephants okay what is the guarantee that such a thing even exists right do you think such a thing always it should exist there is any reason to suspect that such a thing should exist again I would request you to think about matrices all right think about matrices look at the following set I the identity of of course size n cross n let us say that I mention of this is n okay I a a squared dot or dot until a to the n squared how many objects are sitting inside this set n squared plus 1 does something strike you at this point what are matrices after all can you not write up a matrix as an n squared cross n vector is it not if you think of it as an operator also you capture an operator by its action on what a basis set and each action on a basis element is captured by exactly n bits of information if you see the first element of the basis v1 look at the action of the operator phi on v1 so phi v1 now phi v1 is equal to some summation alpha i times vi is again right. So, every action on phi vi is captured by n variables and there are n such vi's so it is again n squared numbers essentially at the end of the day so the entire information about the action of an operator on a vector on an n dimensional vector space is captured by n squared bits of data so think of this matrices as n squared cross 1 tuples how many such n squared cross 1 tuples are here n squared plus 1 so if in terms of Euclidean spaces when viewing matrices as Euclidean spaces it is a bit of imagination not much so you can view this like I have laid this identity out as some 1 0 0 0 0 then the second column so column by column if I sort of laid out it is called a VEC operation sometimes and those of you who have used MATLAB might have come across something like this a VEC operation on a matrix the second one would be 0 1 0 0 0 like this you know last one will be just 0 0 till the last one so this is an n squared cross 1 tuple is it not right each of these by the same token can be written as an n squared cross 1 so the family of operators in fact I think in one of the exercises that we gave you might have had to show this right that the dimension of if I probably proved it in one of the lectures the dimension of the of the space or the vector space of all linear operators from m dimensional vector space to n dimensional vector space is exactly equal to the products of the dimension of the domain and the co domain that is exactly what this is it is an operator so it is n n n squared so in an n squared dimensional vector space I have stacked up n squared plus 1 vectors can they be linearly independent so they must be linearly dependent therefore there exist alpha 0 alpha 1 to alpha n squared not all 0 such that summation alpha i i going from 0 to n squared a to the power i where a to the 0 is of course the identity that is how it is to be understood is going to be identically 0 all right this means that fx given by alpha 0 for that particular choice of alphas alpha 0 plus alpha 1x plus dot dot dot until alpha n squared sorry index should start from 0 you're right thank you exactly so n squared x to the n squared that belongs to this object by definition does it not belongs to the annihilating ideal of a and therefore this is non-empty so at least we are not looking for q vital elephant in a jungle all right so this is non-empty if it's non-empty then it makes sense to search for the generating element after all it's an ideal in the commutative ring of polynomials with identity is it an ideal though just check if you have any polynomial sitting inside here your f a is equal to 0 let's take f1 and f2 to be equal to 0 so of course f1 belongs to let's do it in one shot f1 f2 belongs to the annihilating ideal of a therefore f1 plus f2 together acting on a is equal to f1 a plus f2 a so this is also the 0 operator this is also the 0 operator so that's the 0 operator yeah so the first condition of the of the criteria for an ideal is met so it's a non-empty it's a non-empty set at least and if we now see that it's a it's an ideal as well as we have cleaned then it makes sense to ask for the unique monic generator of this particular ideal okay so what is it next property suppose fx belongs to the annihilating ideal of a yeah that's the that's the that's the smallest degree polynomial that's sitting inside it that's what it was a constructive proof when we showed you that was a constructive proof no I did not just say that it is a it has a it is always it can always be generated by a single element I actually told you that what that single element was that single element was precisely the element with the smallest non-zero degree sitting inside that ideal so suppose fx belongs to this annihilating ideal and qx belongs to fx sorry that's the ring then what can we say about qa times fa the first observation is that although these are matrices and normally matrices don't commute but powers of a commute with each other as we've used earlier also so it doesn't matter whether I write this as fa times qa or qa times fa what should this be it's nothing but this one is just qa times a zero and this one is just zero times qa in either case they're both the zero right so this object also belongs to that set the annihilating ideal of so it's an ideal I already called it an annihilating ideal of a but now you know I put the card before the horse but still I've gone ahead and proved that it is an ideal so that should be reasonably satisfactory then right okay so we'll end this module with the most important definition now that we have convinced ourselves that this is an ideal therefore it is an ideal sitting in because it's an ideal sitting inside the commutative ring of polynomials it must have a unique monic generator it can have multiple generators single generators but when I restrict them to be monic then it becomes unique like we approved so here is the definition the unique monic generator of the annihilating ideal of a is called anyone knows the answer to this what name it's called it's called the minimal the minimal polynomial of denoted by mu away mu subscript a x okay so that's the minimal polynomial it's a definition we know that it has a unique monic generator so let's go ahead and figure out a way to find out this unique monic generator if you can then that is how you get the minimal polynomial of any operator or matrix a right any questions on this so far so I'll clear okay