A ring is a mathematical structure that has two operations, addition and multiplication, and must obey certain axioms. Examples of commutative rings are the real numbers, integers, matrices and functions. An ideal is a subset of a ring which is closed under addition and multiplication. The Hilbert Basis Theorem states that a polynomial ring is an integral domain and any sequence of polynomials can be written as a finite number of polynomials. This allows us to replace analytic functions with polynomials.

A ring is an algebraic structure with an addition and multiplication operation that must obey certain axioms, such as the existence of zero and one elements. Examples of commutative rings include the real numbers, integers, two by two matrices with real elements, and continuous functions from the real numbers to the real numbers. An ideal of a ring is a subset that satisfies two properties: if two elements are in the ideal, the sum of those elements is also in the ideal, and if an element in the ideal is multiplied by any element in the ring, the result is also in the ideal. Every ring contains two specific ideals: the set containing only zero, and the whole ring itself.

An ideal is a subset of a ring closed under addition and multiplication, and in a non-commutative ring there is a difference between left and right ideals. The Hilbert Basis Theorem states that if a ring is an integral domain, then the polynomial ring is also an integral domain. This implies that for a sequence of polynomials, there exists a finite number j such that all polynomials after j can be written in terms of f1 to fj. Studying the quotients, forming a series and studying where it converges and is bounded can be used to replace analytic functions with polynomials.

A ring is a set equipped with two operations, addition and multiplication. The two operations should be associative and commutative, meaning the order of the elements does not matter. Today, the speaker will discuss the Hillwork Basis Theorem, which is a common algebraic concept. It will be used in the upcoming talk to replace analytic functions with equivalent polynomials.

A commutative ring is an algebraic structure with an addition and multiplication operation that must obey certain axioms. These include the existence of an element called zero, which does not change an element when added to it, an element called one, which does not change an element when multiplied by it, and the distributivity of multiplication over addition. Examples of commutative rings include the real numbers, the integers, two by two matrices with real elements, and continuous functions from the real numbers to the real numbers. Polynomial rings are also a type of commutative ring.

An ideal of a ring is a subset that satisfies two properties: if two elements are in the ideal, the sum of those elements is also in the ideal; and if an element in the ideal is multiplied by any element in the ring, the result is also in the ideal. Every ring contains two specific ideals: the set containing only zero, and the whole ring itself. An example of an ideal in a ring of continuous functions is the set of functions that equal zero at the origin.

An ideal is a subset of a ring which is closed under addition and multiplication. In a non-commutative ring, there is a difference between left and right ideals. An example of an ideal is the set of all elements of the form p1 a1 plus b2 a2 plus bn am, where b1 to bn are any element of the ring. An important type of ring is a mytherian ring, where every ideal is finitely generated. An example of this is the real numbers, where the only ideals are the zero set and all of itself.

The Hilbert Basis Theorem states that if a ring is an integral domain, then the polynomial ring is also an integral domain. This implies that a polynomial ring in any finite number of variables is an integral domain. The theorem also implies that for a sequence of polynomials, there exists a finite number j such that all polynomials after j can be written in terms of f1 to fj. Studying the quotients, forming a series and studying where it converges and is bounded is important for replacing analytic functions with polynomials.

oh hey everyone so i'm one of um actually i just my drawing table is a bit going a bit touchy as well i'm just gonna try and restart it but uh yes i dan's master students working on some singular learning theory stuff and so um so yeah so today for the supplementary session i'm just going to go through some background algebra which will come up a lot in next week's talk uh and so the the focus of next week's talk will be about trying to replace analytic functions with uh equivalent polynomials and so to sort of understand what's going there uh some of the tools we'll use are just rings and uh ideas and most importantly um yeah today i'll talk about the hillwork basis theorem uh so anyone who's taken an algebra course and you will probably be very familiar with this background stuff um so i'll try and i'm gonna try and introduce it in a way that's not necessarily the most formal but try and uh hopefully uh emphasize what the sort of model what the point is of each object or theorem or at least what the point is for the stuff we'll be doing uh so i'll start off just by defining a ring um so everything which i'll usually denote by this sort of r is a set and it's equipped with two operations uh so which are called edition i usually generated by plus and multiplication which uh usually denoted by a dot but sometimes most of the time we just omit that um and in particular these two operations should behave roughly as you would hope so go through specifically what i mean by that but sort of vaguely they should work mostly like they work in the real numbers uh i guess i should say that everyone sort of has their own specific definition of a ring depending on what they're using it for um so sometimes people have different uh extra axioms and other times they have less but i'll just go through the ones which we'll be using um so what the multiplication and addition should satisfy is if you have uh three elements that are arbitrary elements of the set um firstly the addition should be associative meaning that if you add a and b together and then add c that should be the same as adding those b and c together and then a and likewise multiplication should be associative uh the other thing is that edition should be commutative meaning that you can just change the order that you write them in um so i think in general in a ring multiplication [Laughter] uh doesn't need to be continued uh in all the rings that we'll be dealing with though uh today and in next week's talk the multiplication

will turn happy to turn out to be commutative uh okay sorry my drawing tablet only lets me use sort of the left two thirds of the board but i'll try and make it work uh and also uh so if you think about what happens in say the real numbers or the integers uh there should be an element which we call zero which does the same thing that zero does in for numbers so specifically if you add anything to zero you should just not change that element and there should be an element uh called one which does what one does in the real numbers say that if you multiply something by one that should not change it and then a couple more things uh for each element a that should be a strong letter there should be an element which i call minus a which um makes when you add them together you get zero uh i'll just go over to the next form to test out the last axiom um the last thing is that the multiplication and additions should sort of work well nicely together which just means that distributivity should hold so if you add b and c together and multiply by a that should be the same thing as if you just multiply through the brackets like that uh and i should hold whether you multiply a on the left or right so just going through some simple examples of rings um so i guess i already mentioned briefly the real numbers with its normal addition and multiplication forms of ring itself size all those things and similarly uh so does the integers um because yeah if you multiply two integers together you get back another integer i'll just briefly mention like an example of a ring where the multiplication isn't commutative would be uh two by two matrices with real elements because um matrix multiplication doesn't in general commute uh so for a slightly different example we could continue could consider all the continuous functions from the real numbers to the real numbers and so multiplication and addition in this ring would be defined point wise meaning that if you have two continuous functions uh then f1 plus f2 would be the function that sends x to f1 of x plus f2 of x uh and similarly for the multiplication it's defined in the exact same way so the main type of ring that um i've been using in my thesis is polynomial rings um so if you've seen so if you've taken an algebra course you've probably seen this defined in a fairly more formal way um i think so the most uh i guess easiest way to the quickest way to define it is to say like given a commutative ring meaning that the multiplication

um commutes the polynomial [Laughter] in one variable it is written as just the ring and then x in square brackets and that's the set of polynomial functions which go from the ring to the ring again so uh an example of an element would be uh for example a0 plus a1 x plus a2 x squared where these things a0 a1 and a2 are the coefficients and they all come from the original ring r um so i'm going to go run over to the next board and so you can similarly have polynomial rings in as many variables as you want so for example you could have a pointer numbering with three variables um which is the set of polynomial functions from the ring cubed back to the ring um so when starting rings the question that comes up like if you're given a ring uh how can you study its study or understand um its structure how can you sort of understand what it looks like and how it behaves uh i guess yeah in a lot of maths uh the way you try to understand that object which is uh or try to understand some sort of set is to look at all of its subsets so for example you've taken group theory um you've seen that an important way to understand a group is to uh try and find all the subgroups and so similar for rings um and so in particular you we look at a certain type of subset which has specific properties and those are called ideals so ideal of r is this offset i oops which um satisfies the two following properties uh the first one is that if you have two elements in i then adding them together should also be and the second one is that if you take any element in i and then any element not just the vibe of the whole ring um if you multiply them together that should keep you in i so every ring that there's two [Music] so there's two specific ideas that appear in every ring uh specifically the set which is just this contains zero because if you add zero to itself you get zero again and multiplying anything by zero gives you zero and the other uh ideal that's in every ring is the whole ring itself ah very ideal that looks slightly different could be in the set of continuous functions from r to r uh where the idea could you could define an ideal as the set of functions which are equal zero at zero because if you add two of these functions together the result will equal zero zero and also if you multiply one of those by any other continuous function it will still equal zero at the origin um just a quick question yeah these are are these are you considering just commutative rings or are these just are

they left ideals or right yes i guess yeah i should yeah i should say so um and yeah i guess for everything that i'll talk about i'll just consider immediate rings um mainly because that's all i end up dealing with in my thesis and uh then don't again also then don't have to uh worry about the specifics too much oh to make it clear to everyone i was referring to the r a multiplication yeah might be different different if it wasn't commutative yeah yeah that's a good point um so a yes specifically what i've written out would be a left ideal meaning that when you multiply on the left by anything else you stand the ideal and so in a non-commutative ring there's a difference between left ideals and right ideals and right ideals are basically the same definition but they sort of close by multiplication on the right yeah so just go over to the next chord and give a sort of important type of example uh which is if you have a finite set of elements like a1 to a n in the ring you can define what's called the ideal generated by these elements a1 to a n and so that's just written by putting those symbols in these pointy brackets and so that's the set of all elements of the form p1 a1 plus b2 a2 plus bn am where b1 to bn or just any element of the ring uh and so it's not too hard to say that if you add two elements of that form together you get a third element of the same form and also if you multiply one of those by anything else in the ring power it still has that form uh so a specific example of one of these uh would be in the integers uh the ideal generated by three is uh set like m times three given ends in the integers and so that's also you could think of that as this set of all elements that oh integers that three divides um so when you come across an ideal the question that uh is often important to ask is um does it have can you write it as an ideal that's generated by a finite set um and so in a certain type of written so i guess yeah so then there's types of rings where every ideal is finitely generated or has a finite generated set and that's what's called a mytherian ring so uh write it up that so an ethereal ring ah it is a ring where every ideal uh is finitely january so uh yes and ethereum rings uh pretty important and knowing that a ring is ethereum tells you a lot about it um so [Music] uh i guess a important example is just the real numbers uh and that's because the only ideals of the real numbers are just the zero set and all of our self which can be written as

the ideal generated by one and so why is that um that's because say if you have an ideal that's not just zero then you can take an element a which isn't zero and then one on a is in r so being an ideal means that one and a times a which is just one you see an i and so then from that it follows all of the i is all of r and so that this is something that by the exact same argument holds for uh any field um or a field is just a ring where you can invert the multiplication um and so i'll just run back to the first board to introduce the main thing let's talk which is the basis theorem and so the hillar basic theorem tells you when a polynomial ring is an ethereum um okay so that so what the theorem says is that if a ring uh is an ethereal [Laughter] uh then the polynomial ring is an ethereum and so uh this then sort of a short argument automatically implies that the polynomial ring in any finite number of variables it's an ethereal as well [Laughter] um so yeah i'm not going to prove the serum uh it's actually the first not too difficult though so if anyone's interested uh yeah i'd recommend just checking the wikipedia was a couple of proofs um and they're yeah not too difficult to follow um but in particular what this implies is that uh if you have a sequence of like an infinite sequence of polynomials um in this polynomial [Laughter] that um there exists some finite number j so that the ideal generated um by all of all of the fj's are all the infinitely many fjs uh is equal to the ideal generated by just this finite set of the first capital j of those polynomials and in particular that means that a role and greater than this j you can find uh polynomials uh which i'll write a j n [Laughter] um i should use and to not conflict with the other and um so that the that fm so any polynomial after capital j can be written uh in terms of the of f1 to fj uh in particular in this film and so yeah so what else be doing in the main talk next week is studying what these what these i guess what are called quotients the amj's studying what they are and because there's infinitely many of them you can form a series out of it and studying where that series converges and is bounded will be important for trying to replace these these analytic functions with polynomials um and that yeah this whole basis theorem will come up a lot in that argument um yeah and so that's all i have uh so if anyone if anyone has any questions about any of that yeah feel free to ask um yeah if nobody has any questions i'll let