Fields

Now we've had a gentle introduction to abstract mathematics with groups, we're ready to explore a slightly more complex structure: Fields.

Just like groups, fields have a set of elements that can be combined. However, this time they can be combined with not one, but two operations. One operation, $\cdot$, behaves like multiplication and the other, $+$, behaves like addition.

These two operations must obey the following properties:

Closure

Very simple one we already saw in groups. Combining any 2 elements from the field under any operation should produce another member of the field.

Associativity

Another one we've seen already from groups. The brackets must behave the same way with both operations.

Identity

There must exist an additive and multiplicative identity in the field. The elements $0$ and $1$ are the additive and multiplicative identities, respectively. The decision to label them as $0$ and $1$ is arbitrary, but it is a common convention.

Inverse

Every element must have both an additive and multiplicative inverse. The elements $-a$ and $a^{-1}$ are the additive and multiplicative inverses, respectively. Again the decision to label them as such is arbitrary, but a common convention.

Note the special case defined here for multiplication. The additive inverse doesn't need to have a multiplicative inverse. This makes sense becase there's nothing we can multiply by 0 to get 1.

Commutativity

We should get the same result from either operation even when we swap the order of the elements.

Remeber that this wasn't a strict requirement for groups, but it will be in fields.

Distributivity

This is a completely new rule that we didn't have with groups. This tells us how the 2 operations interact when used together.

Now we've defined our rules. Let's take a look at an example of a field!

Integers modulo 3 $\mathbb{Z}_3$

This is like the set of integers, but with a twist. We only care about the remainders when divided by 3. So, we have the following elements:

We can think about this as counting on a clock with 3 hours, starting from 0:

Say we want to check what $2 + 2$ is. Normally,$2 + 2 = 4$, but we don't have 4 in our set of elements. So we'll need another way to think about addition.

With normal addition, to calculate $2 + 2$, we'd point to the number $2$ on the number line and then move 2 places further along. Let's do that but using our clock:

So in our field, $2 + 2 = 1$! Earlier we said the elements of our field were the integers remainder when divided by 3, so this result makes sense since 4 divided by 3 gives a remainder of 1. This gives us another way to calculate results here. Instead of using the clock, we can just do the calculation on the normal number line and then take the remainder of the result when divided by 3.

And multiplication just behaves as we would expect. Let's draw out the addition and multiplication tables:

So does this satisfy our rules? Let's check!

• Closure
  • We can see every cell of each table is also an element of the field so this is clearly true.
• Associativity
  • This one's not hard to prove, but it takes a lot of work since we'd need to check every combination. Here's one example: $2 \cdot (1 \cdot 0) = 2 \cdot 0 = (2 \cdot 1) \cdot 0$
• Identity
  • From the addition table we can see that $0$ is the additive identity and from the multiplication table we can see that $1$ is the multiplicative identity. This is because they both have the property that when combined with any other element, they don't change the element.
• Inverse
  • We now know the additive identity is $0$, so for each element we need to find another element that when added to it gives $0$. We can easily see this in our table $1 + 2 = 0$ which means 1 and 2 are eachother's inverse respectively.
  • For multiplication, we know the identity is $1$ and we can see that$1 \cdot 1 = 1$ and $2 \cdot 2 = 1$ so 1 and 2 are each their own multiplicative inverse.
• Commutativity
  • There's a very nice visual way we can prove this. If we draw out a line from the top left of the table to the bottom right, you will notice that the table is symmetrical along this line. This means that the order of the elements doesn't matter.
• Distributivity
  • Like associativity, this one is a bit of a pain to prove. We'd need to check every combination. Here's one example: $2 \cdot (1 + 0) = 2 \cdot 1 = (2 \cdot 1) + 0 = (2 \cdot 1) + (2 \cdot 0)$

So we've proved that $\mathbb{Z}_3$ is a valid field!

Although we have only a few rules on fields, there are actually many interesting properties that we can prove from these rules. We'll do one example here to give a flavour and then leave some more to the exercises.

One of the things we know from basic maths is that if $a \cdot b = 0$ then either $a = 0$ or $b = 0$. Here's how we can prove this using the rules above:

We'll do a proof by contradiction. We'll assume the opposite is true and then show that it implies a contradiction. So we'll assume $a \neq 0$ and $b \neq 0$. Then we have:

Since $a \neq 0$ we know it must have a multiplicative inverse so we can multiply both sides by it to get:

This violates our assumption! An equivalent argument could be made for $a$. So we have proved our statement!

Move on to the exercises to see if you can prove some more interesting facts!

Exercises

Exercise 1

Exercise 2

Exercise 3