 Alright, thank you. Okay, so let me start with the screen share. Okay, so. Right, so today's lecture is going to be right so we've been talking about quadratic forms over over the past past couple of lectures. And today we're going to sort of, I guess have a little bit of a reset. And this will be no quadratic forms today. So instead we will talk about the ball. So I will introduce the the pediatric numbers, which will play a central role in the rest of the rest of these, these three weeks. Okay, so, yeah, so today's lecture is going to be about the pediatric numbers and some of its properties. So, the idea of the pediatric numbers is that, so they're going to be built. So there are there are sort of number system that will be built in a in a way that you're supposed to think of as analogous to the construction of the real numbers. So, well, although I, I guess I said that we, you know, I guess I said quadratic forms are going to play a central role today let's just go back to our original motivation which is we want to talk about. So we want to know when we can solve a quadratic equation when when when so when when is a quadratic form isotropic over the rational numbers. And one of the obstructions is, you know, it needs to be it needs to be isotropic over over the real numbers which is conditional in science. So, the idea is that we're going to get a bunch of other obstructions which come at each prime P from the pediatric numbers and together they're going to be enough. So in this in mind let's let's first start by recalling the construction of the real numbers. Right, so so somehow the ideas of this construction of the pediatric numbers is sort of parallel to the construction of the real numbers, and a lot of concepts that we can sort of do over the real numbers are going to have sort of analogs in in the in the pediatric world. But, ah, sorry, there is a chat. Okay, well, I will keep going. Yeah, so let's recall the construction of the real numbers. So, yeah, so, you know, you've seen the construction of the real numbers for example via Dedekin cuts. But actually for this talk will be convenient to use the construction of the acoustic sequences, because I can cuts rely on the ordering of the real numbers but we're not going to have in the setting so so recall the construction of the real numbers, an element of our is an equivalence class of course she sequences x by where these are rational numbers. And so it's a course she sequence meaning that the differences x i minus x j go to zero as i and j go to infinity. So maybe I should write that security sequences, ie, x i minus x j goes to zero as i comma j go to infinity. And well it's an equivalence class I need to specify the equivalence relation. So, so here, a sequence x by and a sequence y sub i are equivalent so they define the same real number. So the equivalence x sub i minus y sub i goes to zero as i goes to infinity. So, in particular is defined as a sort of a completion of the real numbers where we sort of add in formal limits of course she sequences, and we mod out by the natural equivalence relation. Sorry, so there's. There's a lot of echo. Oh, okay. Is that a problem with my audio. Sorry, the reason for echo is because I'm giving oral instructions in so cocoa to people to switch to a different zoom and some people are in a different zoom. Sorry. Okay. Okay. Okay, thank you. All right. So I'm going to continue. Okay, so right so it's possible to sort of axiomatize this somehow more generally. And so yeah so we'll use the following definition. So, let K be a field. And then an absolute value, right, so an absolute value on a field K is a function. So just use the usual absolute value sign, and it goes from K to the non negative reels. And it has the following properties. So one is that you have the triangle and quality so absolute value of x plus y is less than or equal to the absolute value of x plus y. It's multiplicative. So the absolute value of x times y is the absolute value of x times the absolute value of y. And finally, the absolute value of x is non negative real number, and it's, it's zero if and only if x is equal to zero. So this is a notion of an absolute value on on a field K and right so, for example, we have the usual absolute value on the rational numbers. Okay. So if you're given an absolute value on a field K, then the field K acquires a structure of a metric space. So given an absolute value, the field K becomes a metric space. So where the distance between two elements x y is the absolute value of their difference. And, right, so we're going to say that a field with an absolute value is complete. So we're going to say that K is complete. If this metric space is complete. And right so what does it. So that means that every Cauchy sequence converges. And so now whenever we have a field with an absolute value. We have the following general construction. So given a field. K with an absolute value. So then there's always, so it might not be complete to begin with, but there's always a way of sort of enlarging it to make it complete. And then form the completion, which is a larger field so it's called this completion. K hat. So, which is an extension of K. So it's an extension of K and the absolute value. So the absolute value on K also extends to K hat, which is complete. And in fact it's the minimal extension of K together with its absolute value which is, which is complete. Okay. So this is defined well this is defined. I guess it's defined as the completion of the metric space that the underlying metric space of the field K. So K hat is explicitly constructed as a metric space completion. So in other words, as equivalence causes of Cauchy sequences of K. So for example, if we use the usual absolute value on the, on the rational numbers and the real numbers with the completion with respect to this absolute value. So for example, sorry. So the real numbers are the completion of Q with the usual absolute value. And so you might object that this definition is maybe a little bit is a tiny bit circular because right now we defined it absolute value is something that takes values and the real numbers. So we have one is defined make using this is the definition one should be a little bit more careful how one sets it up but I'm going to assume we have the real numbers to begin with. So we won't worry too much about that. Okay, so, but in general the ideas of this construction of the completion of a field with an absolute value is is some sort of generalization of the usual of the construction of the real numbers via Cauchy sequences. So maybe let me give one more example. Sorry, so there's a question from left to real. Yeah, just go ahead. Yeah, thank you I wanted to ask whether just to make sure if I remember things correctly. So this completion if we have a valued field K. Now that is unique up to isometric isomorphism right so any other conclusion is not just isometric but also isomorphic as fields. Yes. Oh, great. Yes. Yes, it's completely yeah it's completely canonical. Yeah. Yeah, I mean like the completion of the metrics. Okay. Right so maybe just another example I want to give is. Right, so you can define an absolute value so let's consider the rational numbers where you've added a square root of minus one secure join I, and define an absolute value such that the absolute value of a plus I times B is a square plus B square. So, sorry, because I should say square root of a square B square. So this is an absolute value. And the completion is going to be given by the complex numbers. Right, so, so I define this notion of absolute value. And I guess the right so so the point is that the key observation is that right so we can get the real numbers and the complex numbers using this construction. But but the key observation is that there are a lot, there are many more examples of absolute values on the rational numbers. So we have the key observation. There are many more examples of absolute values on the rational numbers so we have the usual absolute value function. But, but there are lots of other examples, and that sort of satisfy the same axioms, and there's well okay this first of all there's a very boring example, which is the trivial absolute value. Define an absolute value and this actually works on any field so define absolute value such as the absolute value of x is equal to one, if x is not zero and zero of x equals zero. So this is going to write so this is going to give you a discrete to apologize your discrete metric space on on the field or this works on any field. Okay, so this is not so interesting but we should mention it. Right, but so in fact on the rational numbers there, they're going to be a whole bunch of absolute values, which one for for each prime number. So let me make the following definition. So let P be a prime number. So, right, so given a rational number so given rational number x q. So I'm going to define the pediatric valuation. So, which I'm going to general by word sub p effects. This is called the pediatric valuation. Well it's going to be the number of factors of P that occurs in x. So it's, well if you write x as a fraction, it's the number of factors of P in the numerator. It's the number of factors in the denominator. So in other words, if x is equal to, you can always write x is some power of P, some power of P, P to the r times m divided by n where m and n or co prime to P, then the pediatric valuation is equal to r. So, or, or P of x is some integer and and by convention we're going to say that or P of zero is equal to infinity. So or P is a function from rational numbers to the integers union infinity, called the pediatric valuation. Right so so so this pediatric valuation is going to satisfy the following properties. Right, so first is that the pediatric valuation of x times y is equal to the pediatric valuation of x plus the pediatric valuation of y by, I guess by unique factorization. The pediatric valuation of x plus y is going to be greater than or equal to the minimum of the pediatric valuation of x and the pediatric valuation of y. That's because, well if x is divisible by P to the r and y is divisible divisible by P to the s then x plus y is divisible by P to the minimum of r and s. Right and so finally, well, so essentially sort of by convention, the pediatric valuation of x is equal to infinity if and only if x is equal to zero. So now we can define the pediatric so that's a pediatric valuation. And because of these properties what we can do is define the following so we're going to define the pediatric absolute value of x to be p to the negative of the pediatric valuation of x. So this is the pediatric absolute value. So this pediatric absolute value is a function that has the property that well, it's a so roughly it's small it's going to be small, if and only if x is divisible by a high power of p sort of by construction the pediatric valuation is large if you have a number which is divisible by high power of p which means that this pediatric absolute value is small. And so if we look at the properties one two and three above, they translate naturally into properties of the pediatric absolute value. So first of all we have that the absolute value the pediatric absolute value of x times y is the absolute value of x times the absolute value of y. Second we have that the pediatric absolute value of x plus y is at most the maximum of the pediatric absolute value of x and the pediatric absolute value of y. And finally, the pediatric absolute value of x is equal to zero, if and only if x is equal to zero. So this just comes from translating the properties of the panic valuation. And now we observe that these are well these are almost exactly the properties needed to ensure that you have an absolute value. So these are close to the axioms of an absolute value, except that, instead of the triangle inequality. We have something that's actually a lot stronger, which is. Of the triangle inequality. We have this property that the absolute value of x plus y is less than or equal to the maximum, instead of the sum of the absolute value of x and the absolute value of y. So in particular, so this is stronger so in particular the pediatric absolute value is an absolute value function. But in fact it satisfies this even stronger property, which is called a non Archimedean property. So this is, so this is indeed an absolute value, but it satisfies this additional condition which is called being non Archimedean. Right so so for example, I mean, this is saying that if you have any integer, it's pediatric absolute value is is at most one. So for example the pediatric absolute value is less than or equal to one on the integer C. And I mean where is it large it's large on something like one over p to the 1000. So, right, so we have the following definition, which is that the patic numbers is defined as a completion of of the rational numbers with respect to the p patic absolute value. So in particular the patic absolute, the patic numbers are themselves have an absolute value so so this patic absolute value extends canonically to QP. And QP is complete with respect to it in other words Koshy sequences automatically converge. So it's a completion as a as a field with absolute value of the rational numbers with respect to the patic with the patic numbers. So this is entirely analogous to the construction of the real numbers. As the completion with respect to the usual absolute. Right, so I guess maybe we should think a little bit about. So, you know, how do we think about elements of the patic numbers. Well if you think about, you know, how do you think about an element of the real numbers. Well you can abstractly define it as an equivalence class of Koshy sequences. Koshy sequences of rational numbers so so that's sort of Koshy with respect to the, the Archimedean absolute value, and sort of one one sort of example of this is that you could have a decimal expansion. That gives you a real number, because you can think of sort of the truncations of the decimal expansion is giving you a bunch of rational numbers, and those former Koshy sequence in with respect to the usual absolute value and the limit is as well it's the infinite decimal expansion that's the associated real number. So, in the real numbers, we can represent any real number via a decimal expansion. And right so just, I mean I just want to emphasize so so so this completion it's an element of the completion, you can think of it as an equivalence class of course I mean it is defined to be an equivalence class of Koshy sequences. So an element of QP is, well, by definition, it's a, it's a, it's a Koshy sequence of rational numbers Koshy with respect to the pediatric absolute value, and that's defined up to equivalence, but we want to make things a little bit sort of more explicit So when we think about the real numbers often we think about a decimal expansion, which, in particular is a way of representing a real number as a Koshy sequence of rational, namely the truncations of the decimal expansion. So there's something entirely sort of analogous to that over the pediatric numbers. So right so proposition. So any pediatric number has a unique pediatric expansion of the form. It's a sum from I, well I is going to start with some some negative integer. And then it's going to go all the way to infinity. And it's a sum of a sub i times P to the I, where a sub i ranges over 01 dot dot dot through P minus one. So given any pediatric. So right so given any pediatric number you can you can write it in a pediatric expansion. Well, in sort of an expansion of base P in this form. So note that right so, so note that this is different from the usual sort of decimal expansion or binary expansion if you prefer of a real number. So note that the expansion goes in the opposite direction. Sorry, so there was a question. Yes. So when I say I greater than or well I write so this means that I is starting at minus and for some, some enlarge so so it's an it's an infinite some, but it, it starts, it doesn't go, it's not infinite to negative infinity. So that helps. I mean kind of like a decimal expansion right I mean it, it, it's finite in one direction. Sorry, there's a question from Garrett. Yes. So, just like with the decimal versus binary and the real numbers, can we help us some sort of base change in the pediatric numbers. In terms of that like powered some like instead of like doing it with respect to P with respect to some other number P prime. Yeah, so right so maybe I should. Yes, maybe I should say some things about. So this is a problem. I'll say something or the proof that I should say some things I should try to sort of compare and contrast what happens with the real numbers. Right so over the real numbers. Yeah so right so you can you can do a decimal expansion but you can also do a binary expansion or a turnery expansion and so forth. So the numbers, well, you have to be a little bit careful because. So what what does first of all what does this even mean this means that you're taking an infinite sum. And the infinite some has to. I mean you're supposed to think of that as a limit of finite sums, which, you know is giving you some sort of koshi sequence. And for that, what you need is that you need the terms to go to zero, if the infinite some is going to converge. So in fact, sort of a fun consequence of the non our community property is that for an infinite sum to converge and the patic setting it just suffices it's necessary and sufficient for the terms to go to zero. But, so, for example, so what does this infinite some mean it means that it's, you know you're taking the sum of the ace of items p to the I, but the point is that that's going to zero those terms are going to zero because he has absolute value less than one. Right, so maybe I can add this to the note this converges. This even makes sense to begin with, because, well the absolute value of P is full. It's one over P. So in fact for what I'm saying right now it's not, you can sort of normalize the patic absolute value, however you choose the key thing is that the absolute value of P is less than one. So that that's why this actually converges. But on the other hand if you were to choose a different prime number. So if you were to choose some prime number q, then the absolute value of q would be equal to one. So this wouldn't converge. I mean if you if you wrote it if you tried to expand patic numbers q adequately, it wouldn't, it wouldn't converge anymore. So what you could do is you could try to for example you could expand it p squared adequately and so forth, but you have to choose. Okay, so this converges because the absolute value of P is is equal to one over P. So this this infinite sum actually sort of makes sense. So that's why also it goes in the in the opposite, the opposite direction as the decimal expansion decimal expansion converges because one over 10 is less than one, they are comedian setting in the, but here P is less than the absolute value P is less than one. Okay, so so right so it goes in the opposite direction, it's need to use base P. And you could also use like base P square I guess you could use any basis to display people usually one just use base P. So maybe one other point that's actually kind of important is that the petic so the decimal expansion is not unique of a real number, because point nine repeating is equal to one for example, but the petic expansion is actually unique. So, so in that sense the petic expansion is actually it's it's more canonical in the decimal expansion, because it's it's really, it's really unique. Okay. Right, so, so how you know so why is, why is something like this true. Well, right. So, so unlike the decimal expansion. Right, so how do you produce a petic expansion. Right, so, for example, let's, let's first say so so if you if you have a petic, a petic number you can think of it as a Koshy sequence of rational numbers that are getting closer and closer to each other in the petic topology. So suppose, given a right so suppose you're given a petic number it's a yields a Koshy sequence of rational numbers in the petic topology. And yeah I mean maybe let's let me just make things even sort of simpler for myself by assuming that you have a sequence of integers. So what you do is well any integer has a finite petic expansion, finite base p expansion. And if you have a sequence of integers that are Koshy and the petic topology, then these base p expansions, they're going to sort of stabilize in any given range of digits. So, I mean of course that the base p expansion can, it can go off to positive infinity, but in any finite range it's going to stabilize and it's going to have no negative terms. So in this sequence, the collection of base p expansions stabilizes, and that that is, well it's going to stabilize in any finite range. And the limit is going to be the, the petic expansion, the limit is going to be an infinite expansion of the associated petic number. Okay, and so right so in general you, you're allowed to have some right so in general you might be working with rational numbers. You can do something kind of like this so if you have a rational number you can always find something some finite expansion and base p, such that it's periodically very close to it to to any rational number. And if you have a sequence of rational numbers that form something Koshy and the petic topology, then that's, that's going to end up converging. Okay so maybe just as an example. One over one minus p is going to equal one plus p plus p squared plus p cubed. So this doesn't make sense over over the rational number this doesn't make sense with the real numbers but it does make sense in the petic. Okay. So this is a sketch of, of, of why this is why this, why this fact is true and this is roughly how you should, you know, this is roughly how you can think about, like sort of representing a petic number which is that it's, you know, it has some sort of petic, kind of expansion. So as you've been saying, you should sort of think of the petic numbers is in some way analogous to the real numbers. Because right it's defined with respect to petic numbers are defined as a completion of the rational numbers, but instead of with respect to the usual absolute value with respect to the petic. So, I want to discuss some of sort of the comparisons between petic numbers and real numbers. There are some similarities, but also some sort of dramatic differences. And one similarity is that they're both locally compact. So both. Well so both of these are, I mean, both of these have absolute values so in particular they, they give you metric spaces and logical spaces, and both of them are locally compact. So if you have a closed disc of some finite radius, then in the real numbers and in the petic numbers. These are, these are going to be compact. And this is really useful this means that for example if you have any bounded sequence it has a convergent subsequence. And I mean this is one of the things that's going to. It's going to make it it's going to make it a lot easier to solve equations in the petic numbers or in the real numbers then, then over over the rational numbers. So both are both are examples of, of locally compact, locally compact fields. But now let me maybe say difference. So a difference is that the petic numbers are, are, are non our comedian, which means that QP contains a sub ring, given by ZP, which is all those elements X of QP. Such that the absolute value of X is less than or equal to one. So in other words, the unit desk and QP. So the unit disk is so equivalently it's going to be those petic expansions, going from zero to infinity, instead of from minus 10 to infinity. Right so this is the difference between the petic numbers and the real numbers because in the real numbers if you take the unit desk it's not a sub ring it's not close under addition. But because QP is non our comedian, if you have two petic numbers whose absolute values less than one, then their, their sum is also is also going to be is also going to be less than or equal to one. So that's going to make ZP into into actually a commutative ring. And, well, so ZP is a commutative ring. And in fact, one way of thinking about an element of ZP says I said ZP you can think of it as it's those petic expansions that start from zero and go to infinity instead from from minus 10 or something. And you can think of an element of ZP as a compatible system of congruence classes mod p to the end for each end. So more formally it's the inverse limit, inverse limit of Z mod p to the end. And so what that means is that it's compatible systems of congruence classes mod p to the end for each end. All right, so there's no way of producing sort of an analogous sub ring in the real numbers. But in the petic numbers you have this nice sub ring of, of the disk at radius one. Okay, so this is a fundamental difference. Another fundamental difference is, yeah if you think sort of topologically, so the real numbers are connected. So I guess if you think about, for example, if you think about real numbers via data can cuts I mean this is really sort of fundamental that you build the real numbers, so that they're connected. So the question what is an inverse limit sorry I guess this is, this is a definition. So maybe I won't define inverse limits in general, but a pediatric integer can be specified as a system of congruence classes modulo p to the end for each end. So I mean if you look at like the first end digits of the petic expansion that's determining the congruence class mod, I guess my p to the n plus one repeat at the end. And so as as you sort of allow an infinite expansion that that means you're allowing a compatible system of congruence classes modulo p to the end for for each end. Okay. So, sorry, are is connected. So let me explain another fundamental difference which is that yeah if you think sort of topologically are is connected. And, well the pediatric numbers are not connected. So the pediatric numbers are not only not connected I guess they're what it's called totally disconnected. But maybe instead of trying to define. Well let me just say that what so in fact I think it may suppose a little bit on the problem set so the pediatric numbers they it looks like. So if you look at any disk in the pediatric numbers, it's it's both open and closed and it looks like it looks like a cancer set. So, for example, if you look at the ZP which is the unit that's the unit desk in QP. Then this is both open and closed and in QP. Right so I guess if you have a connected space you don't have not allowed to have many open and closed subsets. So it's both open and closed so that's again this is a consequence of really the non Archimedean property. Yeah, so it's really a consequence of the non Archimedean property, because when you say ZP is the unit disk. It's the unit disk I guess centered at, at zero but you could actually center it at any point of the unit desk so the non Archimedean property. And that means that it's, it's, it's a yes. Yes. Thanks. Sorry, the question was definition of totally disconnected. In fact, ZP is is homeomorphic to a cancer. Right and so I mean that's because you can, is EP is sort of an element of ZP is is literally the same thing as a pediatric expansion. Well, a pediatric expansion starting and starting at zero, which is some sort of. Whereas a real number is or say a real number of the unit interval is is given by a decimal expansion but you've also sort of. You're modded out but you're portioned by some equivalence relation on those decimal expansions. And so it might not sound like a lot but somehow topologically they look very different the unit interval is this sort of nice connected thing. And ZP is maybe a little bit more unfamiliar. It's, it's more like a country, it is literally homeomorphic to a cancer site. So, I mean that's that's wonderful in many ways but it's, it's also different. It takes some time to get used to. So let me also mention sort of one general difference so right so this is somehow the world of non Archimedean fields instead of the world of Archimedean absolute values. And in fact, if you work with Archimedean absolute values so an Archimedean absolute values mean something which is not non Archimedean. In fact, the only complete are complete Archimedean fields are the real numbers and the complex numbers. But in fact you have tons of fields which are complete with respect to. So lots of complete non Archimedean fields. So in particular if you start with the real numbers it's not I mean it's not algebraically close but it's, once you add a square root of minus one it becomes algebraically closed. But over the pediatric numbers that's no longer true there are lots of ways you can enlarge the pediatric numbers and get sort of larger examples of fields which are complete with respect to a complete non Archimedean absolute value. So, lots to extend to complete non Archimedean fields. So one example is that QP has, I mean the algebraic closure of QP is not finite over QP so it's not like the complex numbers you just add a square root of minus one. But any finite extension of QP is going to have is going to be a complete non Archimedean field as well. So you can also, you can also find even larger much larger constructions, the sort of no size limit the sort of no size limit in in the complete non Archimedean case there's a size limit in the Archimedean case you can only, you can't get any larger than the complex numbers. Okay, so I guess bearing, you know, these facts in mind so there are some similarities and also some very salient differences. Somehow the principle, one principle is that you should think about the real numbers and the pediatric pediatric numbers as being sort of on a similar footing for, at least for some questions. So, for some questions. Well, for example the questions involving quadratic forms that are going to be discussed in this course. One can regard the real numbers, and all the pediatric numbers on sort of a similar footing and sort of consider them together somehow. Right so so maybe let me give a little bit of motivation for why, why you might consider this. Well, in fact, there's a theorem of Ostrowski. So, I think this is on the problem set which is already in SoCoCo. So the theorem of Ostrowski is that any absolute value on the real numbers, I'm sorry on the real numbers on the on the rational numbers is well up to equivalence, it is the Archimedean absolute value or it's the So any non trivial absolute value on Q is equivalent to a pediatric absolute value or the Archimedean absolute value which I'm maybe I'm going to denote as so the usual one to know with a infinity subscript. So the claim is that this is actually a list of all of them, and I should just say what equivalent means, and equivalence is sort of a question of normalization. So as I mentioned in the definition of the pediatric absolute value. It didn't really matter so actually convenient for something in the future but it'll be, you can you can you can also define the absolute value of P to be, say, one half, instead of one over P. And it would still give you the same topology would still give you the same notion of convergence and if you complete the same notion of completion. So this means here. The same up to raising to a power. And if you have, if you have equivalent absolute values then that doesn't, it's not going to change the notion of completion or the topology and so forth. Right, so, so there's this complete classification of all absolute values on the rational numbers they're either pediatric for some crime P, or the rational numbers. And so, yeah, so somehow the idea is that you should consider all these completions sort of together. And so this turns out to be sort of a useful, you know, somehow a useful philosophy for, for various purposes. So let me, let me give sort of one example, which is, which is the product formula. So if x is a rational number. Well, I guess x is a non zero rational number. Then the statement is that the product as P ranges over all the prime numbers, and also infinity of the absolute value of x at P. So we're allowed it we're also including the Archimedes value is equal to one. And so I guess there's no convergence issue so almost all the factors are one. And so there's definitely many factors are, are one. And so the center product makes sense. There's no convergence issue and the product is equal to one. So, this is saying that yeah you have all these different petic absolute values and somehow when you combine them you, well you get this nice formula. And so this is this is one in this case it's not hard to prove because you can, for example, you can use unique factorization and you can check it for a prime number P. And it's going to fall out on tributions and, and it's going to fall out. But there are also lots of, you know, there are also other instances of this type of, of this type of philosophy. So later, so probably in Justin's lectures. We'll see that quadratic reciprocity can be interpreted can be formulated as as a product form, as sort of a similar product formula. So it's going to be a product overall primes and infinity and each of the products that each P is going to be somehow defined on the patic numbers or on the real numbers of P is infinity. So yeah so this is sort of, and so this is sort of an, this is sort of nice because when you formulate it in that way also it generalizes very nicely to, for example, other number fields. And in fact even to function fields so to find an extensions of like Fp, the run series of rational functions over Fp. So, I guess I'll be a little bit about that in the problem set. Um, and in fact one of the main. So one of the main theorems I guess of the, of the mini course is, is the theorem of Hasan Minkowski. Maybe I will just formulate now. So a quadratic form. Well it has a non trivial root or I guess is isotropic. So quadratic form over Q, if and only if it's isotropic over QP for each P and over the real numbers. So this is called a, yeah, and what maybe let me also say to quadratic forms of the, in fact it's a consequence of this fact but to quadratic forms for Q are isomorphic. If and only if they are isomorphic over each QP and overall. So I guess this is called a, well this is called a local to global principle it's it's saying that if you want to solve a problem. Well this particular type of problem in quadratic forms over the rational numbers, then it's suffices to solve the problem over the each prime P and over the real numbers. And it turns out so I guess, probably I will explain this next time. It's, it turns out it's pretty it's much easier to think about these types of questions over, over the real numbers we know that we can do it by looking at signs. And similarly over the piano numbers we can somehow answer these questions by looking at conferences. So this is going to be one of the main, main results of of the course. And I guess. Yeah, so I guess, I guess, it's a bit of not well, so I guess next time what I'm going to do is, I will, I will try to continue with the pediatric numbers so so I will try to continue with the pediatric numbers and in particular will try to find out how you can solve equations over, over the pediatric numbers, using, just using something called Hensel's lemma. Yeah, so essentially what, so you start to explore this a little bit on the homework, but well right so for example over the, when you, in general when you complete, like when you complete from the rational numbers to the real numbers, you add lots of solutions. I mean, there are lots of equations that you can solve in the rational numbers that you can solve in the real numbers. For example, you don't have a square root of two and the rational numbers we have real numbers. And similarly when you sort of complete in the pediatric sense, you're adding solutions to lots of lots of equations. So if you try to determine what, what things you're adding, it's, you have to somehow look at congruences and that's, that's given by Hensel's, Hensel's lemma, so I guess we'll explore that next time. Okay, so I think it's going to be a good place to stop for today. So, yeah, thanks. I'll stick around for a little bit for, for questions. Yes. Okay, so, what, what are some results in real analysis that depend on local compactness. Well, right, so I guess local compactness. Yeah, so maybe let me formulate, so let me formulate a, yeah, let me formulate a result in that vein. So I mean if you think about, so right so what is local compactness giving you it, for example it's giving you that if you have a sequence of real sub sequences, then there's always a convergent subsequence. And right. So, I guess maybe we could get the following example so. So, example. So given a polynomial, polynomial equation f of x equals zero, where f of x is in z brackets x and actually it could be in many variables if you want. So suppose you can solve f of x is congruent to zero, modulo p to the end for each end. So suppose you don't know if you can solve this equation f of x equals zero in the integers, but let's suppose you can solve this equation, modulo any power of p. So, again, the integers. And then you can solve it in, in, in, so then you can solve f of x equals zero in, in the pediatric integers. And in fact it's an if and only if. So, so right so why is that I mean it's, if you have a solution, modulo p to the end for each end, well, I mean you can, you will let's say you have an integer solution modulo p to the end for each end, then you get some sequence of integers. And there is, it doesn't have to converge petically but there's some subsequence that converges petically by local compactness. And so some subsequence that has to converge in the petic topology. And that is going to give you a solution of fx equals zero in, in, in the pediatric integers and conversely if you have a solution in the pediatric integers well then you can approximate that modulo any power of p by an integer. Yeah, so for example that's something that that local compactness is giving you. Right, so what we'll see next time is, so hence Osama is going to give you a criterion so. So in principle, this is telling you you know how do you solve an equation and in qp or let's just say zp for simplicity, but you need to be able to solve an equation of the integers modulo any congruence. And, well in principle that's an infinite number of conditions to check. And what hence Osama which I will talk about next time gives you is that in fact you only need to. So let's get one do some, like some finite power of p or in fact just mod p if your derivative is not visible by p and then you can automatically sort of get a petic solution. Great. Thank you. And I'm going to my office right now. Okay, if there are any more questions I can. I'm going to stop by the office hours. Well, if not, I guess we'll, we'll see you tomorrow or we're in office hours and promise that's already a question. Actually I kill, can I ask you a question about yesterday's property. At the beginning, you, let me go find the page first. Yeah, at the beginning of yesterday you mentioned about any n dimensional quadratic form over F q sub q is isomorphic to one comma one comma one D. I wonder if this is for an isometric form, or just any quadratic form. Sorry, any, any I some, sorry, there's isometric to something that one. So, so any, any quadratic form can be has this a normal form that you can come close into the power of the hyperbolic and plus a v prime which is and I saw, I saw, I saw tropic. And I saw, I don't know how to read and well I mean I think sorry over over the yeah so over the over a finite field. The normal form is, so you can always. Yeah, so you can, you can always write it as a direct sum of copies of, of like one, I mean you can always diagonalize it, but in fact you can always diagonalize it whether it's all plus ones except for one term. So, so this is for this is for general is not just for and, and, and I saw tropic form to even for any form. So what I'm wondering is, so in general we write hyperbolic form as one comma negative one. So, here, suppose I have a so what is the hyperbolic plan. If it is a hyperbolic plan, would that be one comma one or one comma negative one I'm a little confused. So in general the hyperbolic plan over any field is given by one comma minus one. I mean so it's the hyperbolic plan is the form x squared minus y squared, for example, or you could also write it as the form x y. So what about if it is for F sub q, then would it be one comma one or one common. So, so, so that statement should still be true even for hyperbolic form, right. So, so what do I expect about the D. Okay, so if you have a, if you have a hyperbolic form just a one. So if you're the hyperbolic plane, then it's one comma minus one so it's already in that form. It's already in the normal form that you're looking for. But then, then, then that means that the so that means that all the previous ones that's going to be the portion for the v prime for the end isotropic portion. So not quite because, well, so the key feature of sorry so maybe they're so they're two different sort of normal forms that maybe are coming up here so what there's a normal form that you can do over any field. Well, I mean, over any field a quadratic form you can decompose it into an anisotropic plus a bunch of copies of the hyperbolic plane. Right. Yeah. So that's something you can do over any field. And over over the, and the anisotropic part is sort of uniquely determined. In, if you're over a finite field any form of dimension at least three is automatically isotropic. So what you can, what you can do in over a finite field or over any C, C one field or any field of you and very intense to. So you can write any form as a direct sum of copy of the hyperbolic plane plus a two dimensional or plus an utmost two dimensional anisotropic form. But this was actually a different normal form that I was, I think at the beginning of yesterday's lecture, which is that you can always write as a bunch of copies of plus one, and then at most one term, which is not plus one. So there's a different. So I'm very confused now so suppose I have just one comma one all the way to one. So is this Do we know or not know that whether it's, it's an isotropic or not, just by the phone. Well so once, once you have at least three terms is always isotropic. If you're over a finite field. That's true because it's dimension three so it has to be isotropic. So even though if it's 111 that will still be isotropic. So you have to even one comma one can be isotropic so it depends on minus whether minus one square in your field. I see, I see right and it depends on the soul, if you've it is telling you if this is true that it also depends on the basis you choose right because you always have if it is hyperbolic you have to choose the basis, you want to do to such that each of them is isotropic. So if you have a hyperbolic form or hyperbolic plane you can choose such a basis that's right. Okay, so so so just because they're I write it as a one comma one. That doesn't mean that is an isotropic. No, no, not at all. I mean. I mean in general one comma one is right so if you have the form if you have a two dimensional if you have one comma one, then if you're in a field where minus one is a sum of two squares. Or sorry if you're in a field where minus one is a square, then that's always isomorphic to one comma minus one in which case it's hyperbolic plane by rescaling but if minus one is not a square then it's an isotropic. Okay, okay. I'll just get a little bit better. Thank you. Sure. Actually, any, any more questions or basically on what on what you were saying so basically for example if we had a, if we had the one one, you know, diagonal diagonalized the binary quadratic form over f five, where minus one is the same thing as four so it's a square, then that's a hyper that's a hyperbolic plane right. That's right. I see. Thank you. Bye bye. See you tomorrow.