 Welcome back to our lecture series Math 4230, Abstract Outdoor 2 for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Mistletide. Lecture 19 is going to return to a topic that we have talked about already in Math 4230, as well as in Abstract Outdoor 1, Math 4220, which is the idea of polynomial rings. Now with both of the previous exposures we've had to polynomial rings, the coverage was actually very light and it was meant just as an introduction to a new type of ring that we can create from an already well-known ring. In this exposure, we are going to be much more in depth in our coverage of rings and in particular connected to our unit about domains, factorization and integral domains that we've been doing already. So by the end of lecture 19, we'll actually discover under the right circumstances a polynomial ring actually is a Euclidean domain. We will see that actually in the second half of this lecture in a different video. So in this video, what we are going to do is we're going to review the ideas of a polynomial ring and actually specify some details. I should say, you know, go through some details that we omitted beforehand in our lecture series here. Now we won't go through every single detail whatsoever. I would definitely recommend referencing a good algebraic textbook like Tom Judson's, his abstract algebra book, abstract algebra theory and applications. That's a good book where you can find all the details of the things I'm going to skip over right now. I'm doing this, of course, for the sake of time. So what is a polynomial ring? Let's remind ourselves what that is. So first of all, to define and work with polynomials, we need to have a ring of coefficients. So the ring itself needs to be, we have two operations of addition and multiplication. With respect to addition, it's going to be an abelian group. We can add in an associative, commutative manner. There's a zero element. There are additive inverses. And from that, we can infer the notion of subtraction. But we also, for our coefficients, need multiplication. We can multiply together two coefficients. That multiplication will be associative and we will have a multiplicative identity, which we call the unity or the one element of the ring. And there should be a distributive law. There should be the distributive laws between multiplication and addition. Now, we are going to assume our ring is commutative, but very little of any of the theory that we develop about polynomial rings actually requires the coefficient ring to be commutative. But for the most part, you can assume that R is a commutative ring unless stated otherwise. So you have this ring R and then you're going to introduce a quote unquote symbol, which is not an element of the ring, okay? And this element we're going to call an indeterminate. So first of all, what do you mean by a symbol? What actually is a symbol? I want you to be aware that the word symbol here really just means that it's a set. It's a set in the sense of ZFC set theory. In this framework, every element of a set is likewise equal to a set. So if we wanted to dive deep into set theory, we can even talk about the number zero is a set, right? It's really the empty set. It doesn't contain anything. The number one is actually the set that just contains zero, which as zero is the empty set. You have the set that contains nothing. It doesn't literally contain nothing. It contains the empty set. You can define the number two to be the set that contains one and zero, which that means this will be the set that contains the empty set and the set that contains the empty set like so. We can define three to be the set that contains zero, one and two, which of course, then if we go through all the details of that, you have the empty set. You have the empty set inside of a set. You have the set that contains the empty set and the empty set itself. I think that's enough right there. And we can keep on going from there, right? My point is in the set theoretic sense, so in ZFC set theory, everything is a set. And so when I talk about a symbol, we're just talking about a set. Now, I should also mention that as the ZFC axioms prevent a set from being an element of itself, this is, you know, this because we don't want to run to things like Russell's Paradox. We could, we could let X actually equal the set R itself. So from a set theoretic point of view, that's perfectly okay. To use R directly leads to a little bit of awkward notation, much like the awkwardness of the set theoretic natural numbers. We just listed a moment ago. It's kind of this recursive manner. If you're not used to the set there, it can be kind of a weird thing. So alternatively, the word symbol is used here. Intuitively, X is just a symbol that doesn't correspond algebraically with anything in the set. So we can try to go into deep, deep set theory to talk about this. But the idea that's going to be sufficient for us is that this set X is not an element of R. It's something else. It could be the set R itself. It could be whatever. The idea is how do we know there's anything else outside of R? You know, we don't have to worry about that philosophical question, that axiomatic question. So X is itself going to be a something that's not an R. And we typically call it this indeterminate. What does an indeterminate mean? Well, in previous like non-abstract algebra, algebra classes, I guess you call it, you know, concrete algebra, you know, probably like real algebra. If you're talking like a college algebra class or something. When you work with polynomials, this symbol X is often described as a variable, right? And it's called a variable because as the name suggests, it is able to vary. That is, X is just a placeholder for a number that'll be assigned to it later on because maybe X is the input variable of some function. When we work with polynomial rings, we don't take this meaning for X. X is not a variable X. It's already it's already a number. What is that number? Well, it's probably not a real number. It's probably not an integer. It's probably not a complex number because like I said, this symbol X is something that doesn't belong to the ring. So if your ring was like the complex number, C can't be a complex number. It's not a placeholder for a number that's going to put in there. It's just something else. And I get, I get into the set theoretic rant what's going on here, but the idea is X is just a new number that doesn't belong to the coefficient ring and therefore has no algebraic relation to the original ring. And that's why we do prefer this term indeterminate. It's not, you can't determine what X is by R alone. It transcends the ring R. So with that sort of again set theoretic situation now behind us, we won't talk more about it in this lecture series. We can then define the set R adjoin X for which this is going to be the set of polynomials whose coefficients come from R. So we get things of the following form. We get a polynomial where a n, a n minus one, a n minus two, all the way down to a one, a zero. These are all numbers that belong to the ring R, the so-called coefficient ring. And as such, these elements we call coefficients. Then there's these numbers zero, one, two, three, n minus one, and these are all natural numbers. And so we have this linear combination of various powers of this indeterminate symbol X where the coefficients in these linear combinations come from our ring here R. And so this is the then going to be the ring of polynomials. We might call it the ring of R polynomials if we need to emphasize what the coefficient ring is in that situation. Now we often will borrow notation we use from college algebra or calculus or things like that. We often will denote a polynomial by f of X because that's that's how we're accustomed to from like our calculus days for which we can write this in an expanded form like we did before. Sometimes when it's convenient, we'll write it in this more abridged format where we use sigma notation to suggest that we have this sum and then an arbitrary element in that sum will be a monomial because after all that's where the word polynomial comes from in the first place. Poly to suggest several or many. So we have mini-nomials. You know a monomial is just one-nomial. The the gnome here is short for name so that you can think of it as the X is indeterminate. We have many indeterminates as a monomial which just has the one indeterminates just the power of X with the coefficient and I keep on using this word but the numbers that show up here that come from the ring R we call these the coefficients of the polynomial f. Again much like with the function notation we've seen before we sometimes go back and forth between f and f of X. In calculus then of course is it there's a big difference between that. When we say f we're referring to the function. When we refer to f of X we refer to the output of the function with the input X. Because we have an indeterminate element we're not going to fix it on the difference so really we can use f and f of X interchangeably in this abstract algebra sense. But so be aware we're going to do both of these. These numbers a's are the coefficients of the polynomial. We should say of course that the coefficients of the polynomial determine the polynomial. Two polynomials are equal if and only if they have the same coefficients for every power of X. Given any polynomial it only have all but finitely many zero coefficients. That is there's going to be some maximum power n such that any power of X larger than that has a zero coefficient. That largest number n with a non-zero coefficient is called the degree of the polynomial. We'll often call it D E G of f short for degree. And that will work for every non-zero polynomial. If we talk about the zero polynomial the zero polynomial would be the polynomial for which every coefficient is zero. It technically doesn't have a degree because we want the degree of our polynomial rings to be a norm of an integral domain for which you don't have you don't define the norm of the zero element. But there are some situations where maybe we do want to talk about the norm of zero and in that situation we're going to call the the degree of the zero polynomial negative infinity. It's not a real number and it's smaller than every every natural number whatsoever. So that this is mostly just so that formulas are a little bit more consistent. But don't worry about it too much. So in in your polynomial f there is this one monomial one term in there that obtains the largest power aka the degree of the polynomial. That monomial that obtains the largest power we refer to that as the leading term of the polynomial. Its coefficient is called the leading coefficient. The the term that has x to the zero power sometimes called the constant term. If the leading coefficient is one we refer to this as a monoc polynomial and monoc polynomials are going to be polynomials we're going to like a lot as we go into the future. So be aware of this as we are considering these polynomial rings. And so again I want to emphasize before we get off this slide here that this number x this indeterminate element x. x has algebraically speaking no relationships with any of the numbers from r. And so this this polynomial is equal to zero if and only if each and every one of these coefficients is equal to zero. That's the only way it can be. Also there's again there's no algebraic no relations here that x to the n equals x to the m if and only if n equals m. So there's no possibility that different monomials could agree with each other. The only the only algebraic relations that we allow on the number x are the ones that are required to be a ring. So things like associativity, distributive laws, and commutivity will be the only the only relations we require so that it becomes a ring and we'll be we'll be very explicit about the operations of a polynomial ring in just a second. I should say that we do allow it we we actually do allow that multiplication between rings the ring coefficients and the indeterminate they can communicate with each other. So we do allow that but nothing else again other than those ones we require for our binary operations. So what are the operations of a polynomial ring? After all r of x we keep on calling it a polynomial ring it needs to be a ring so it needs addition it needs multiplication. How do we do these things? So we define addition between two polynomials f and g which be aware that the the polynomial f is just going to be the sum of monomials of the form ai times x to the i and b will be the sum of monomials of the form bi times x to the i right here and we can this number and that their degrees don't have to be the same because there's only finitely many non-zero coefficients in this what we can do is we can just tack on a bunch of zero coefficients until they match up so without the loss of generality we can assume these sums have the same number of terms in them. When we add together two polynomials what we do is we end up just adding together the coefficients of the corresponding powers of x. And this is what we refer to as a formal sum of the polynomials although if you're in a college algebra setting you probably would call this combined like terms. What does it mean to combine like terms? What means that you add together those terms which have the same powers of x in play here and why do we do that right? Well maybe we know this maybe we don't but when it comes to so to speak like terms let's say that we have a times x to the m and we add that to b times x to the m well because this is going to be a ring this ring has to have the distributive laws and you'll notice that both of these terms because they're like terms they have the same power of x well if this is to be a ring then I have to be able to distribute that is I should be able to use the distributive laws which means I can factor out the common divisor of x to the m here and so by the distributive law this is the same thing as a plus b times x to the m right here so this is called a formal sum because every sum in every ring no matter which ring you're talking about because it's a ring because it has the distributive law you can combine like terms but because x is an indeterminate that has no other algebraic relations on it other than those imposed by the rings axioms then nothing more can be required by addition other than combining like terms and that's what we mean by the sum is formal it's just combined like terms and nothing else similar vocabulary is what we're going to use to define polynomial multiplication that is we define polynomial multiplication to be formal products so we multiply together two polynomials f times g same thing as before f is some sum with coefficients in a b g is the sum of coefficients column bj right in this situation i'm not necessarily going to suppose that they have the same number of terms their degrees could be totally different from each other not a big deal whatsoever but of course if you added a bunch of zero coefficients to the smaller one or whatever that's not going to make any difference in this sum whatsoever either so the product of two polynomials we define to likewise be a polynomial where now the sum our index will be k k is going to range from zero up to n plus m in which case we have a coefficient ck times x to the k how do you define ck well ck it's itself going to be a sum of products you're going to take the you're going to take together all the possible products of an ai times a bj such that the indices i plus j are equal to the index k like so this is often referred to in the literature as the convolution product between between like vectors what have you and really the convolution product here this this operation here is just exactly what the coefficient the kth coefficient is going to be when it comes to polynomial multiplication well why is that well let's remember the classic foil rule right if i have the product of two things a plus b times c plus d so if you have two so to speak binomials here by the distributive property you can distribute the first term onto both of the terms in the sum so you end up with a plus b times c plus a plus b times d but then using the distributive law again you can distribute the c and the d and you end up with ac plus bc plus ad plus bd i didn't even use the commutative property here this is just a consequence of the distributive laws so the typical foil method works in any single ring you have because it's a consequence of left and right distributive laws all right and so as we know from you know college algebra and calculus and and you know previous classes previous math studies that probably weighed before abstract algebra because of the distributive laws we can foil and we can also do sort of like an extended foil right it doesn't have to be two terms we can have some type of like a1 a2 a3 you know all the way down to some an and you can multiply that by some b1 by some b2 all the way down to some bm and we can multiply this out also using the distributive laws well what if we make this into a polynomial we have x x squared x cubed x to the n like this one we could have x x squared x to the m that's not going to change anything by the distributive laws we can multiply these things out but when it comes to these situations like with the foil method let me erase some of these things on the screen real quick what if we were to rewrite this using some polynomials if we had something like x squared plus 2x plus 1 and you times that by 3x plus 5 this extended foil method which is just the distributive laws they come into play here and you get things like 3x cubed which i'm gonna i'm gonna even slow down on that one we'd end up with x squared times 3x then we end up with an x squared times 5 next we're gonna end up with a 2x times a 3x then we're gonna get a 2x times a 5 and then carrying it down here you're gonna get a 1 times a 3x and then you're gonna get a 1 times 5 so if we have a ring we have the distributive properties and so multiplication of these two polynomials has to do this by the distributive law but then when we multiply these things together we said that coefficients commute with the x's so you end up with 3 times x squared times x you're then going to have a 5x squared like so you can pass the x by the 3 we're going to multiply coefficients together we'll do that one just a second though 2 times 3 we get x times x we're gonna have a 2 times 5 times x you're gonna have a 1 times 3 times x and then finally a 1 times 5 so again because the the indeterminate commutes with the there with the coefficients we get this so as these are multiplications these are products inside of a ring we can simplify those but how do you deal with something like this x squared after all just means x times x you have another x right here because we're associative the exponential laws are going to apply here and so you have to be able to simplify the product of these two monomials you add together their powers so you get 3x cubed we end up a 5x squared we end up with then a 6x squared we're then going to have a 10x we're going to have a 3x and we have a 5 but like we said with addition you have additions associative we have to be able to combine like terms this is a formal sum so you get 3x squared plus 11x excuse me 3x cubed 11x squared plus 13x plus 5 so what we're what are we have to try to say here is that when you do a product of polynomials the axioms of a ring force it to be this thing and so unless there was an additional algebraic relation on the on the number x there I couldn't simplify it any more than that because I don't have those relations you have to be able to combine this and this is just the formal rule for that situation like if we're asking ourselves how did you produce 11x squared well to get 11x squared you have to look for coefficients that'll add up to b2 so like you can take the coefficient one so this is a1 you can take the coefficient here which is b1 so you get a1 times b1 that was one possibility you can also take a2 times that by b0 all right a2 and b0 the sum of those indices work there there is no there is no b2 over here I should say it's coefficient zero so that's the only other combinations you could get a1 times b1 was equal to two times three a2 times b0 that's equal to one times five so we had six plus five which is the 11 so this formal multiplication is what it has to be by the ring axioms anything else we could put on it would depend it would require some type of algebraic relations which we don't have that's the thing I keep on emphasizing in this in this video because there's no algebraic relations on the indeterminate x addition and multiplication are purely formalized so there are a lot more details that we could provide for rings and I'm not going to leave it really I'm not going to do it I'm going to leave it as an exercise for the for the viewer here so some important takeaways here that if r is itself a ring then the r adjoined x is going to be a ring itself the polynomial ring this ring the ring r will be commutative if and only if rx is commutative because the indeterminate element does commute with every coefficient if there's any blockage on commutivity it happens on the r side so r is commutative if and only if rx is commutative one important thing to remember about this polynomial ring is that the coefficient ring itself can naturally be viewed as a subring with unity here because the constant polynomials the constant polynomials of this ring can be identified with the coefficients elements themselves so if a ring has a non commutative subring then it's not commutative so clearly if if r of x is commutative then the subring r is commutative but likewise if the coefficient ring is commutative then everything will commute because coefficients and indeterminates already commute with each other likewise the coefficient ring it's a ring with unity if and only if the polynomial ring is a ring with unity okay so the unity element acts like well clearly it's the unity for r but it will also act like the unity for every polynomial because of that formal product we talked about on the previous slide multiplying that by one will be the unity for everything and for reasons that we're not going to go so much into this video if you have a polynomial ring with a unity it has to be the constant it has to be a constant polynomial and that polynomial has to since it's since it's just a coefficient at that moment has to be the unity of the ring itself one can make a degree argument which will talk about degrees in the very next video of this lecture here additionally if you think about the polynomial ring versus this right here this remember is a set this is the set of functions of the form r to the r there's a natural ring homomorphism from the polynomial ring to the function ring r to the r which again in the previous video we had talked about how this is in fact a ring and if you don't recall the idea is you can define because you have this function from r to r you can define the sum of two functions where if you add together f f of x and g of x in this essence i'm now talking about the image of the function because what does what does the function f plus g do to the element x it's going to map it to f of x plus g of x which is this is the image of x under f this is the image of x under g so you can you can add together functions in the usual sense you can multiply together functions in the usual calculus sense f of x times g of x okay now when we think of polynomials we often think of them as functions which we're treating them now as this purely algebraic object the reason we can do that is because of this ring homomorphism there's a ring homomorphism that sends a polynomial to a function and it's just going to be the associated polynomial function this is in fact a ring homomorphism in particular the composition of this ring homomorphism with the function evaluation which is you take everything in this ring and you can evaluate it for a specific number so f then is mapped over to some evaluation like f of a like so this is likewise a ring homomorphism this would be a ring homomorphism from r to the r to r itself if you compose that with the polynomial ring then this is going to be polynomial evaluation the so-called evaluation map and this gives us a ring homomorphism from the ring of polynomials to the coefficient ring and this will be dependent upon some fixed element like little r inside of r like so these are some very important some very important properties with regard to polynomial rings