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let me introduce you to a strange number the square root of zero some of you might say that is clearly zero but less reserved to judgment a bit other numbers get to have two square roots so why not zero so let's imagine this is a numbers distinct from zero and let's call it the square root of zero how can we imagine this number well we can imagine that it is a really small number an infinitesimally small number a number so tiny that it's square is actually zero this makes sense because normally really small numbers get even smaller when we square them we can then imagine that the square root of zero is that sum I said some level of tiny that we actually care about it but its square is so small we don't and call it zero so we have this intuition that square root of zero is some really tiny number so what can we do with really tiny numbers well calculus is just really tiny numbers let's try to derive something using the square root of zero normally it will derive using this limit let's remove the limit and replace the epsilon with the square root of zero let's have F be this the square function let's compute the value of this expression we get 2x this is the derivative of x squared let's try with X cubed we get 3x squared in fact this works for any power function that is functional so take X to some natural number power since for instance the exponential function can be written as a sum of power functions using the Taylor polynomial formalism we can even do this to the exponential function if you compute this we will actually get the exponential function which is its derivative with this we're doing calculus without any limits note that there are some finer points we are ignoring here such as how division by the square root of 0 only works in certain cases and that X must be a real number from now on since square root of zero is a little long let's call this number Epsilon so what can we do with epsilon well we can multiply it with any real number and we can add real numbers to it from this we can see that we get numbers of the form a plus B epsilon where a and B are real numbers these numbers again can be added in the way that we would if epsilon for a variable and they can be multiplied using the distributive rule and that epsilon squared is equal to zero we recognize the structure this is a ring we have a set of elements and we can add and multiply them if you don't know what every ring is watch my previous video or you can try to google it this ring of Epsilon together with real numbers is called the dual numbers the dual numbers are also called are extended by epsilon modulo epsilon squared we will explain in detail what all of this means first of all are extended by epsilon is the ring of real polynomials using the variable epsilon taking modular epsilon squared essentially means that in this ring we set epsilon squared equal to 0 and continue computing as they would be polynomials but set in zero instead of epsilon squared whenever it arises including when it arises in epsilon cubed or epsilon to the fourth power this is similar to how complex numbers are defined they can be called are extended by I modulo I square squared plus 1 here we said I squared plus 1 equal to 0 which is the same as setting I squared equal to minus 1 the fine details of our ring modulo something is defined where we leave for later but it essentially builds on something called ideals and Co sets of those ideals all you really need to know is that when whatever comes after the modulo is set to 0 and all arithmetic builds on that another important ring defined in this manner is the Gaussian integers these are the integers but including the imaginary unit I the restraint can be called said extended by I modulo I squared plus one another this another similar example is said extended by square root of two were set extended by X modulo x squared minus two these are the integers together with the square root of two we could also do Q extended by square root of two which is the same but with rational numbers instead of integers note that these rings are actually very different from their base counterpart as for instance with the Gaussian integers we have five equal to two minus I times two plus I so five is no longer a prime number but other combinations of integers and the imaginary unit can be in general we can do our extended by X modulo P X for any ring R and any polynomial px with coefficients in R in fact we can also have multiple variables such as are extended by x and y modulo X times y minus 1 in this ring we have X plus y squared equal to x squared plus y squared plus 2 not to tear the x and the y are not elements of art extended by x and y with specific elements x and y in even further generality we can do any ring or modulo any element of our so these polynomial rings are created by exactly the same formalism a set modulo 5 it is worth noting that when it comes to addition are extended by X modulo P of X looks like a vector space over R with dimension the same as the degree of P in particular if R has a finite amount of elements s then are extended by X modulo P has a finite amount of elements s to the D where D is the degree of the polynomial P we will now try to explain a set of very important rings called the finite fields to begin we must first understand what a field is a field is a commutative ring such that every nonzero element has a multiplicative inverse that is for each number X not including zero there is another element Y such that X times y equals 1 this translates into that division is always possible examples or fields are the rational numbers the real numbers and the complex numbers the pierrick numbers are also a field as are the algebraic numbers the integers are not the field since division isn't always possible that modulo n is a field if an only if n is a prime number otherwise it is just a commutative ring so for instance inside mod 5 3 is invertible because 3 times 2 is 1 and it turns out that all numbers inside modulo 5 are invertible in this manner a finite field is then a field which only consists of a finite number of elements indeed we can classify these fields by exactly how many elements they have and it turns out for their given size all finite fields of that size are s morphic since there are isomorphic we can think of them as essentially the same and so all the fields of size and go by the same name F N or G of F n for Galois field as they are sometimes called however there are some sizes for which there are no finite fields for instance 6 there is no field which has size six we do however know already know one example of a finite field and that is set modulo P where P is a prime number we already discussed that these rings are fields so we know that there are finite fields of size and a prime number but are are there more well earlier in this episode we explore ring extensions so perhaps we can add more elements to set modulo P and get other sizes well let's look at set modulo P extended by X modulo of polynomial F when is this a field well it turns out that when we begin with the field and modulo out and a reducible polynomial we get a new field in a reducible polynomial is a polynomial which cannot be factored in the ring that we are working with of course all polynomials can be factored but if we restrict ourselves to the real numbers for instance the polynomial x squared plus 1 cannot be factored and is thus irreducible so let me let's begin with set modulo P and extend by X and modulo al are irreducible polynomial f of degree D we now know that this is a new field but what size is it well the elements are polynomial and take the form a 0 plus a 1 X plus a 2 x squared and so on up to D where each a n is an element and in said modulo P so we have d numbers for which there are P choices of value this is P to the D choices in total so we see that we have P to the D elements in subs modulo P extended by X modulo F and since this is a field when you have a field of size P to the D so we can see that this is any prime not power all that remains to show is that there is always a reducible polynomial of any degree d we won't show this in this video but it is in read-through for instance to get a finite field of size 4 we begin with such modulo 2 and extend by a variable X and modulo at x squared plus X plus 1 which is a known degree to irreducible polynomial it turns out that there are no more finite and those with prime power sighs thank you for the attention I hope you learned something and leave a comment if you have any questions subscribe if you want to see more of this kind of content and I hope to see you again