In abstract algebra, there is a wide variety of operations: geometric transformations,
function composition, matrix multiplication… But many sets of elements have the familiar
operations from arithmetic: addition, subtraction, multiplication and division. Loosely speaking,
if you can add and subtract, you have a group. If you can add, subtract and multiply, you
have a ring. But if you’re lucky and get all four operations, the result is an object
that behaves similarly to the numbers you learned about in arithmetic and algebra. We
call these fields.
One thing we’ve talked about before, but it bears repeating, is that in abstract algebra,
subtraction is actually adding with negatives. For example, “3 minus 5” is the same as
3 plus “negative 5.” So instead of saying “you can add and subtract”, we say “there’s
addition and additive inverses.” Notice that word “additive?” That’s because
there is more than one kind of inverse.
For example, “3 DIVIDED by 5” is the same as 3 times 1/5.
So instead of saying “you can multiply and divide”, in abstract algebra we say “there’s
multiplication and multiplicative inverses.” The additive inverse of 5 is negative 5.
The multiplicative inverse of 5 is 1/5.
So in arithmetic, you learn about addition, subtraction, multiplication and division.
But in abstract algebra, you speak of addition, additive inverses, multiplication, and multiplicative
inverses. It’s a shift in thinking, but it’s key to understanding the more abstract
objects.
We’re now ready to talk about fields. To motivate the definition, we’ll start with
a collection of 6 groups, some of which come with additional features. By adding features
we’re familiar with from the real and complex numbers, we’ll arrive at the full definition
of a field.
Consider these 6 sets: The integers
The 2-by-3 real matrices The 2-by-2 real matrices
The rational numbers The integers mod 5, and
The integers mod 6. Notice that all 6 sets are groups under addition:
They’re closed under addition: you can add two elements together and the sum is in the
set. The negative of each element is in the set.
There’s an additive identity, and the associative property holds.
Better still, all 6 groups are COMMUTATIVE under addition.
So as a first pass, all 6 objects are commutative groups under addition.
The next feature we’d like to include is multiplication.
You can multiply any two integers or rational numbers together.
You can also multiply any two numbers mod N for any N.
That leaves the two sets of matrices. You can multiply two square matrices, but
you cannot multiply 2-by-3 matrices by each other.
Their dimensions are incompatible for multiplication. So only 5 of the 6 sets advance to the next
round of commutative groups with multiplication.
Just as all the groups are commutative under addition, in a field, we’d like multiplication
to be commutative as well. After all, the real and complex numbers are
both commutative, and they are a pleasure to work with.
The integers and rational numbers are both commutative under multiplication, so they
advance. Also, the integers mod N are commutative under
multiplication for any N. But we’re about to lose another candidate.
The 2-by-2 matrices are NOT commutative under multiplication.
There are an infinite number of examples where matrix multiplication is not commutative.
Here’s one example... So only 4 of the 6 are commutative under multiplication.
Next, we’d like each number to have a multiplicative inverse.
The additive inverse 0 is the big exception here.
You cannot divide by 0, so this number cannot have a multiplicative inverse.
But we’d like every NON-ZERO number to have a multiplicative inverse.
Sadly, we’re about to lose two more sets. In the set of integers, only 1 and -1 have
multiplicative inverses. None of the other integers have one.
For example, the inverse of 2 under multiplication is ½, which is not an integer.
And we lose the integers mod 6 as well. 2, 3 and 4 do not have inverses mod 6.
Mod 5, however, is different. Here, every non-zero number has a multiplicative
inverse. You can check this by looking at the multiplication
table for this set. So the only two sets to advance are the rational
numbers and the integers mod 5. By the way, these two sets both have a multiplicative
identity 1. This is not a surprise, since the product
of a number and its multiplicative inverse is 1...
The race is over, and we have two winners. The rational numbers and the integers mod
5. These both share a similar set of properties. They are both commutative groups under addition.
They both have a second operation - multiplication, which makes them rings. Furthermore, multiplication
is commutative, so they are commutative rings. Better still, other than zero, every number
has a multiplicative inverse. It’s this last property which puts them over the top
and turns them into fields. We’re now ready for the textbook definition of a field.
A field is a set of elements “F” with two operations: addition and multiplication.
Under addition, the elements are a commutative group. Under multiplication, the non-zero
elements are a commutative group. And addition and multiplication are linked by the distributive
property. This is the compact definition of a field. If you wanted, you could define a
field and make no mention of groups whatsoever. You could just give a complete list of all
the properties a field must satisfy. This is fine, but you do lose sight of the fact
that a field is actually two groups with two operations at the same time.
Let’s return to the two examples of a field we just saw: the rational numbers and the
integers mod 5. The rational numbers are denoted by “Q”
for “quotient”, since every number in this field is the quotient of two integers.
The rationals are an infinite field, while the integers mod 5 are a finite field.
But the integers mod 5 are not the only finite field.
In fact, the integers mod “P” for ANY prime number “P” is also a field.
Together, these form the starting points for ALL fields.
That is, if you pick ANY field “F”, then it will contain one and only one of these
fields as a subfield. We call these fields “prime fields”, and
say that “F” is an extension field. If “F” is an extension of the integers
mod 2, we say it has “characteristic 2.” If it’s an extension of the integers mod
3, it has “characteristic 3.” And if it’s an extension of the integers
mod P, we say “F” has “characteristic P.”
But if F is an extension of the rational numbers, we say it has “characteristic 0.”
So the characteristic of a field tells us which prime field it extends.
The are an infinite number of fields in mathematics. We begin by learning about the “big 3”:
the rational numbers, the real numbers and the complex numbers. But there’s an infinite
number of infinite fields, and even an infinite number of FINITE fields! From finite fields
to Galois extension fields, you’ll find many uses for this structure.
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