 Welcome back to our lecture series 42-20 Abstract Algebra 1 for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Missildine. This is the first video in lecture 13, which is based upon section 4.2 from Juddson's Abstract Algebra textbook about the multiplicative group of complex numbers. So in this lecture, what I want us to do is review the topics of complex algebra and use this to define a cyclic group that lives inside well actually families of cyclic groups that live inside the complex numbers. And so in order to do that, there is a necessary review of complex numbers and I don't know necessarily how much a review you as a viewer are going to need because there are many places in the standard undergraduate curriculum where complex numbers come up. The most obvious one would be a course like complex variables which by the end of that course you should be quite quite affluent in the language of complex complex numbers. Personally, I use a lot of complex numbers when I teach a class like linear algebra that's math 2270 here at Southern Utah University. You might see trigonometry like in a calculus class I should say you might see complex numbers in like a calculus class trigonometry college algebra. And so if necessary, you're gonna find some videos at the end of this lecture that will review the basic ideas of complex algebra. So the things that you do need to remember right now is that when you see the letter I we're referring to the complex square root of negative one. There's no real number which squares to be negative one. So we invent a new number which we call a complex number in this case imaginary number that does exactly that. Complex numbers are the linear combinations of a real number plus an imaginary number. This first part the real part the second part is called the imaginary part. The thing you really should know about when it comes to complex algebra adding and subtracting complex numbers is essentially just combining like terms. When it comes to multiplication as you have every every complex number is essentially two dimensional. There's two terms there when it comes to multiplying complex numbers really just foil them out. What you really need to know is the following when you have i to the zero this is just equal to one because any nonzero number raised to zero should be one. i to the first is itself when you square i well because it's square negative one this is equal to negative one and when you get i cubed as this will be i squared times i you're gonna get a negative i and then when you were when you go to the next one i to the fourth is gonna give you one because you get negative one squared which is one i to the fifth is equal to five i to the sixth is equal to negative one and i to the seventh is equal to negative i and this pattern just kind of repeats itself over and over and over again powers of i can be computed just by reducing the exponent by four and considering the the residue that is the remainder when you're done with that because every power of i will be one of these four numbers one i negative one negative i so for example if you did like something like i to the 27th power we can recognize that i you'll have i to the 24th which is a multiple of four times i cubed multiples of four we can just ignore because those are just equal to one and so this becomes i cubed which is negative i on the other hand if you take like i to the 101 Dalmatian power there well that would break up as i to the 100 times i to the first 100 of course is a multiple of four four times 25 and so you just end up with i right here so we can compute powers of i by using a reduction of the exponent mod 4 complex division comes down to basically rationalizing the denominator we multiply the top and bottom by the conjugate of the complex number so a plus bi is equal to a minus bi like so and so we can then compute all of all of this are complex arithmetic can be done now additions attraction location division I want to mention though that this anomaly we see about powers of i is kind of a curious thing if we take the set of the four numbers you see right here one negative one i negative i this this together actually forms a subgroup of the multiplicative group of complex numbers because notice one times anything will be that other thing negative one will just switch the signs if you times i if you squared you can negative one if you cube you can negative i if you take the fourth power you get one this this set right here is closed under multiplication it has an identity it has inverses one's its own identity inverse excuse me negative ones also its own inverse multiplicatively and then i and negative i are there are the reciprocals of each other this gives us a subgroup of order four inside of c star and in fact it's a cyclic subgroup what we want to do in this lecture is essentially generalize this cyclic subgroup it turns out these subgroups are quite ubiquitous in the complex field here c star and so now you should hopefully see some links on the screen if you need some more practice with addition subtraction of complex numbers click the link to the top left need some more practice with multiplication complex numbers please click the link to the top right if you need some practice with division of complex numbers click the link to the bottom left and then finally as it is well you know if you're if you feel okay with any of these things you can skip ahead to the next video about primitive roots which is in the bottom right on the screen