 Now I want to introduce also as we're talking about fields right here the idea of a skew field or sometimes called a division ring Because as we see it in group theory commutivity is not required for inverses To have multiplicative inverses you do have to have a multiplicative identity So unity is required you need a ring with unity, but you don't need commutivity So if you have a ring with unity that satisfies the multiplicative inverse axiom every element except for zero has an inverse We call this a division ring or a skew field the idea skew field is that the prefix skew often suggests commutivity That is non commutivity. So there's some type of twist when you do the multiplication Now a skew field could be commutative or could not be commutative the thing is we're not assuming anything about commutivity So fields are examples of division rings because in a division ring We have a well-established addition subtraction multiplication division The the difference of course though is in a skew field or in a division ring We don't assume that multiplication is commutative which also makes division sort of an interesting thing as well Because you get a left division versus a right division But but we won't worry about that so much right here So like the rational numbers the real numbers the complex numbers all examples are commutative division rings, so I think it's worth entertaining. Is there an example of a non commutative Skew field of some kind well It can be difficult to come up with examples because you can actually prove that there's no such thing As a non commutative finite skew field that is every finite division ring is an actually a field But let me give you an example of an infinite one And this is going to be based upon the quaternion group that we studied previously in this lecture series So the quaternion group Which you'll recall consists of eight elements plus or minus one plus minus i plus or minus j and plus or minus k And we can identify these eight elements with with matrices in the general linear group of two by two complex matrices Well, the general linear group of these two by two complex matrices is a subset It's a subset of the ring of two by two complex matrices And this here is a ring first of all, but it's also a vector space If we think that as a complex vector space since it's two by two matrices, we'll get a four-dimensional Vector space, but as every complex number z can be written as a plus bi Where a and b are both real numbers We can actually think of every complex number that is the complex number field itself Is this two-dimensional vector space? And so the two by two complex matrix ring forms an eight dimensional real vector space And so what we're going to do is we're going to construct the the subspace of two by two complex matrices spanned by these four matrices All right So we're going to we're going to take the span as a real vector space of the matrix one i j and k now notice We don't need to include minus one minus i minus j minus k because as the span takes all linear combinations We have negative one times each and every one of those things So if we take the span of the quaternion group, we get the so-called quaternion ring So this is often referred to as the ring of quaternions. Sometimes people call it the hamiltonian ring named after of course hamilton who First discovered the quaternion group in the first place. He was so excited about it He vandalized a bridge with his discovery of of the quaternions. All right So the quaternion group forms a spanning set for this subspace So it's going to be a subspace which means it'll be closed under addition And it'll be closed under skill and multiplication, which just means multiplication by a real number But what's the multiplication of this of this ring? Well, it turns out that because we can we know how to multiply together these quaternion elements Um, it's just going to be the same multiplication of the complex matrices in play And you can show that in fact this is a sub ring of m two by two c That is this this subspace h is in fact closed under multiplication Um, and it's going to be a non commutative multiplication Which m two by two of c is non commutative, but a subset actually could be commutative The issue here is that h here contains the quaternion group as a multiplicatively closed subset Which that subset is non commutative. It's a non-hubilant group So this sub ring is going to be non commutative So this gives us a non commutative ring with unity. It does contain the identity I claim though that it is in fact also a division ring, which we'll talk about how we do the inverse of an element just a second But I want to give you a slightly different perspective how one could think of the quaternions here You could also think of the quaternions as an extension of the complex field Because the reason why we call this symbol a lowercase i is we could think of it as this Complex matrix. Yes, but we could also think about also as the complex unit In c so the square root of negative one But what we're saying is here is that in addition to the usual complex square root of negative one We add in two quaternion complexes Excuse me two quaternion square roots of negative one So these are distinct square roots of negative ones that are not added in inverses of each other i j and k In which they multiply together by the usual rules of the quaternion groups So i times j is k j times i is negative k j times k is i etc um, and so as a quaternion Right, you have a number like a plus bi plus C j plus dk where a b a b c d are real numbers You can think of a quaternion as a generalization of a complex number because complex numbers are just those quaternions Whose j and k parts are zero And so we can extend the notion of the complex conjugate in the following way if we take a quaternion bar That is a plus bi plus c j plus dk if you take the conjugate of that This you'll just switch the sign of the non real parts So you'll switch the sign of the imaginary part i Also the j part and the k part and the and the quaternions there And so then in terms of addition use combined like terms subtraction same thing multiplication Just foil these things out Again that multiplication Just extends that of the complex numbers again We can represent that inside this quaternion inside this matrix ring if we so wanted to but the division Let's get to the division. How is this a skew field a non-communist skew skew field? Well, if I can tell you what the inverse of an element is then we can do division because Division is just multiplication by an inverse So the inverse of a generic quaternion, which it should look like a reciprocal one over the fraction What we're going to do is like we did with complex numbers if you want to compute the reciprocal Just take the reciprocal And then top times the top and bottom by the conjugate of the reciprocal And then what you can then see is that the numerator you're going to get one times the conjugate So the numerator is just going to be the conjugate the denominator is the fascinating part when you multiply out the denominator You're going to end up with This element right here you get a sum of squares and how does that happen? Well, when you do the possible foil here, you're going to get an a squared You're going to get a negative a b i you're going to get a negative a c j and you're going to get a negative a d k That's the first round Then the second round what you're going to get is you're going to get an a b i i'm going to write that here actually you're going to get a You're going to get a b i you're going to get a negative b squared i squared. All right, so i'm going to put that over here You're going to get a negative b squared i squared next you're going to get a negative You're going to get a negative BCIJ. And then next, you're going to get a negative BDKI, like so. Then if we do the next one, CJ, when you distribute it through, you're going to end up with a positive ACJ. Next, you're going to get a negative BCJI. This thing's noncommutative. The order matters here. Next, you're going to get a negative C squared, C squared, J squared. I should have written I squared earlier. And then lastly, you get CJ times negative DK. So you're going to get a negative CDJK, like so. I haven't done that one yet, so I'll put that here. Negative CDJK. And then for the last one, you distribute the DK through, you're going to end up with an ADK, positive ADK. Then we're going to get a negative BDKI, a negative BDKI. How do I get a KI earlier? That should have been an IK. Sorry about that. The order does matter on these things. Then we should get a DK times a negative CJ. So that's going to be a negative CDJKJ, like so. And then lastly, we're going to get a negative D squared, K squared. So there's a lot of stuff to do there, a lot of stuff to do there, but notice how it simplifies out. Some of them are pretty easy. A negative ABI plus ABI to cancel. A negative ACJ cancels. The negative ADK cancels with the positive one. So those are all gone. But then look at these ones. These ones differ only by the terms IJ and JI, which JI is the same thing as negative IJ, so this actually become a positive BC. IJ, so they cancel. Same thing here, KI is the same thing as negative IK. So these terms are going to cancel. And then KJ remembers the same thing as a negative JK. So those will cancel. So everyone canceled off except for this term right here and this term right here. But notice I squared, J squared, and K squared, these are all complex, excuse me, quaternion square roots of negative one. So in the end, this will turn out to be A squared. Then you're going to get a plus B squared plus a C squared plus a D squared, thus giving us what we needed in order to get these units. That is to say that we get multiplicative inverses for any quaternion number.