 So let's see, we were looking at finite fields, trying to figure out all the properties, field, etc. and we made significant progress, we said if somebody says f is a finite field, all the various things we know. Size of s has to be equal to p over m, p is some prime number. Inside ns, what is p? p is you call p as 1 plus 1 plus 1 p times, what should it be? It should be equal to 0 and it should be the smallest one that happens. So p is the characteristic. And we also know that that's also n dimensional vector space over zp. zp or a copy of zp that's inside that vector space. So this is true. And this gives a lot of handle over addition. And it also gives what's called vector representations of the finite field elements. So you know it's a vector space, any n dimensional vector space can be represented using m coordinates, the vector of m coordinates from zp. So every element of the field can be represented as a n coordinate vector with elements from zp. So that's the first representation, it's called the vector representation. This gives the vector representation. And we've been seeing this vector representation as actually a polynomial. So polynomial is also can be thought of as a vector. So if you take the coefficients of the polynomial, collect them together, it's a vector. So vector of polynomial representation. So that also. And the next thing we saw perhaps a bit more surprising, it's the multiplicative structure. We saw that this s has to have an element which is probably called beta or alpha. Beta. Which has this interesting property. It generates the entire multiplicative group. So that is beta power p power m minus 1 is equal to 1. So up until that you will never get any representation. It's a primitive element. Beta is a primitive element. And we also saw this very interesting relationship. I want you to keep this relationship in mind. It's very important. The root of x power p minus 1 minus 1 in s of 1. What are the roots of x power beta power m minus 1 minus 1? All the non-zero elements. So these are all non-zero elements. So this is an important and powerful property. You should keep that in mind. So if you take. So what does this mean? This basically means x power p power m minus 1 minus 1 factors into linear factors in s. Let's see. It can have at best beta m minus 1 roots in s. There is a c. And what are all the roots? 1 beta beta square root. So if you actually do this. x minus 1 times x minus beta. So up until x minus beta power p power m minus 2. What should you get? You should get x power p power m minus 1. So it's a kind of a surprising little property that all the other powers in this product have to cancel. And you have to get x power p power m minus 1 minus 1 in the left hand side. Those are all the roots. So the largest part of x is very easy to check in this formula. x power p power m minus 1 on both sides. You can try to check the constant term. What will be the constant term? There will be a minus 1 power p power m minus 1. And that's going to be positive or negative. It depends on p. In case it won't matter. So the sign will work out properly. Where about the other product? beta power 1 plus beta power 2 plus so on till beta power p power m minus 2. So you can see that there will be a multiple of p power m minus 1. And that will actually be 1 more. So other terms, for instance, if you want to check the coefficient of x. What will be the coefficient of x in this product? What's the coefficient of x? Maybe not x. The x power p power m minus 2. What's the coefficient of x power p power m minus 2? It's what? Minus 1 plus beta plus beta square plus so on till beta power p power m minus 2. Okay, I can't get rid of this. So what should I do? What's the secret? Select the one that doesn't matter. Oh man. That I'll never present for the rest of my life. Just when you think that something happens to you. So will this help? What do I do? Sorry. So F10 in the recording will stop. At least that much I can do. It's frozen or what is the story here? Anyone knows enough about these things too? Somehow nothing is responding. Okay, stop.