 So another important type of algebraic structure is known as a field, and a field is a commutative ring F where every non-zero element is a unit. Remember a unit is a ring element with a multiplicative inverse, so this definition is also the same thing as saying that in a field every non-zero element has an inverse. Now this existence of an inverse for all elements allows us to prove very quickly some of the key properties of a field. So let F be a field if A is not zero, if it's not the additive identity, then its inverse is unique. And so we'll employ a standard process in proving a statement like this. We'll suppose we have two inverses of A. Suppose A inverse and B are both inverses of A. We have AB equal to 1, the multiplicative identity because B is an inverse of A. I can multiply both sides by A inverse, and associativity allows me to group the A inverse A, which is going to give me the identity element. And because one is the identity element, 1 times B is B, and A inverse times 1 is A inverse. And so these two things that are inverses of A have to be the same thing. Another useful property of a field is that for all A, A times zero is equal to zero. And let's consider the product A times B plus zero. The distributive property applies, so this is the same as A times B plus A times zero. But since zero is the additive identity, B plus zero is just B. So our left hand side is just A times B. And since we're in a field, A times B has an additive inverse, so we can eliminate it and get zero is equal to A times zero. Another useful property of fields is that they have no divisors of zero. So suppose we have a product equal to zero, where one of them is not equal to zero. Well then, once again, since A inverse has to exist, since A is not equal to zero, then I can left multiply by A inverse. A inverse times A is the identity, and since the product of something with zero has to be zero, the right hand side is going to be zero. And so if a product is equal to zero, where one factor is not zero, we know the other factor must be. Or we could say that if a product of field element is zero, then at least one of the factors must be zero. And finally, we have one important property. If a field has more than one element, the multiplicative and additive identities are distinct. So suppose we have something that is both an additive and a multiplicative identity. Now if our field has more than one element, there has to be something besides B. So for any A not equal to B, B times the sum A plus B must be B times A, because B is the additive identity, and A plus B has to just be A. On the other hand, we can apply the distributive property B times A plus B is B A plus B B. And since our assumption is that B is the additive identity, we know that B times A is going to be zero. And so B A has to be B B, but since B is the multiplicative identity, that says that A has to be B. But we assume that A is not equal to B. And so this is a contradiction. No, no, that's OK because our assumption was that B was both an additive and a multiplicative identity, and we wanted to prove that it wasn't. So the contradiction tells us that our original assumption must be false. Now if we put this all together, this does lead to some important properties. And one of the most important here is the distinctness of sums and products. And if we have a field, if A plus B is the same as C plus B, then A and C have to be equal. Similarly, as long as B is not equal to zero, if A, B is C, B, then A and C have to be equal. So I'll introduce one term. A Galois field is a field with a finite number of elements. Galois was a 19th century French mathematician who got himself involved in one of the revolutions at the time, wrote up a bunch of papers on what is now called group theory, and managed to get himself killed in a duel at the age of 20. And so now here's something we want to consider. What is required for integer arithmetic mod n to be a finite field? We already know that integer arithmetic mod n is a ring. We also know that if n is prime, then every element has a multiplicative inverse. On the other hand, if n is not prime, then there are elements without multiplicative inverses. And remember, the distinguishing property of a field is that every non-zero element has an inverse. And so that tells us that integer arithmetic mod n is a field if and only if n is prime.