✖

Fields

To follow this explanation you do not need any special maths skills or knowledge. All you need is the very basics of secondary school maths and a willingness to learn something new.

The bogeyman called the algebraic field

If the very name “algebraic field” fills you with dread, rest assured that there is really nothing to be afraid of. A field is in fact a very simple thing, once we look at it from the right angle.

An algebraic field is any arbitrary set plus two binary operations that satisfy certain conditions. That is all. Nothing more and nothing less. As you can see, there is nothing dangerous or terrifying about it. And now that we know this, we can go and take a slightly closer look without any fear.

The set of elements of a field

We will call our arbitrary set T and call its elements the elements of the field. Believe me, friends, the set of elements of a field can be absolutely anything. It could be the set of all even numbers, the set of all your odd socks or even your collection of cuddly toy elephants. Perhaps the only condition is that the set T must not be empty. We would not have much fun otherwise.

Binary operations

If it is not immediately clear what a binary operation actually is, you can picture it as a little mill. You put two elements of some set into it (hence the “binary” in the name), turn the handle gently, and out drops another element of your set. Division is a nice example of a binary operation. Want to know what 10 / 5 is? Take the binary operation of division (the mill), where your set is the set of real numbers. Pour in the ten first and then the five, turn the handle and, lo and behold, out drops a two.

We will call our two operations addition and multiplication, and write them as ⊕ and ⊗. Do not be misled by the names. These operations need not have much in common with the addition and multiplication we normally use. That is precisely because the elements of a field can be any objects at all. And “our” addition and multiplication certainly cannot be applied to cuddly elephants or socks.

Operations on a field

The operations ⊕ and ⊗ must satisfy certain conditions. We cannot introduce them completely arbitrarily. The point of introducing these operations is to generalise the ordinary addition and multiplication that we use on numbers. We want to be able to define the operations so that we can use them on the elements of a field, which, as we already know, can be absolutely any objects, not just numbers. Even so, we want ⊕ and ⊗ to keep some basic sense of addition and multiplication. We ensure this by introducing ten conditions that these operations must satisfy for our structure to be a field. We will call these conditions the axioms of a field. Here they are:

  1. Commutativity of addition – For any two elements a, b of the field T, a ⊕ b = b ⊕ a must hold.
  • Associativity of addition – For any three elements a, b, c of the field T, (a ⊕ b) ⊕ c = a ⊕ (b ⊕ c) must hold.
  • Existence of a zero element – There must be an element of the field T, which we will call 0 (careful, not to be confused with the number 0), with the property that for any element a of the field T, a ⊕ 0 = a.
  • Existence of a negative element – For every element a of the field T there must be an element of the field T, which we will call -a, with the property that a ⊕ −a = 0.
  • Commutativity of multiplication – For any two elements a, b of the field T, a ⊗ b = b ⊗ a must hold.
  • Associativity of multiplication – For any three elements a, b, c of the field T, (a ⊗ b) ⊗ c = a ⊗ (b ⊗ c) must hold.
  • Existence of an identity element – There must be an element of the field T, which we will call 1 (careful, not to be confused with the number 1), with the property that for any element a of the field T, a ⊗ 1 = a.
  • Existence of an inverse element – For every element a of the field T that is different from 0 there must be an element of the field T, which we will call a−1, with the property that a ⊗ a−1 = 1.
  • Distributivity – For any three elements a, b, c of the field T, a ⊗ ( b ⊕ c ) = ( a ⊗ b) ⊕ (a ⊗ c) must hold.
  • Non-triviality – The zero element and the identity element must not be one and the same. That is, 0 ≠ 1

For anyone who would like to practise mathematical formalism, I have written out the axioms of a field here the way they are written in mathematical shorthand.

  1. ∀a,b ∈ T   a ⊕ b = b ⊕ a
  • ∀a,b,c ∈ T   (a ⊕ b) ⊕ c = a ⊕ (b ⊕ c)
  • ∃0 ∈ T   ∀a ∈ T   a ⊕ 0 = a
  • ∀a ∈ T   ∃−a ∈ T   a ⊕ −a = 0
  • ∀a,b ∈ T   a ⊗ b = b ⊗ a
  • ∀a,b,c ∈ T   (a ⊗ b) ⊗ c = a ⊗ (b ⊗ c)
  • ∃1 ∈ T   ∀a ∈ T   a ⊗ 1 = a
  • ∀a ∈ T   ∃a−1 ∈ T, a ≠ 0   a ⊗ a−1 = 1
  • ∀a,b,c ∈ T   a ⊗ ( b ⊕ c ) = ( a ⊗ b) ⊕ (a ⊗ c)
  • 0 ≠ 1

Now you may be thinking that there are terribly many axioms. But all of them are easy to understand once we realise what they mean. The first four describe the usual properties of addition and the next four the usual properties of multiplication, which are also very similar to the first four. The ninth axiom says how addition and multiplication are related. Only the tenth is rather a technical matter, which makes sure that the field cannot have any strange properties.

Notice how neatly we have avoided introducing subtraction and division. These operations are defined by adding the negative element and multiplying by the inverse element. So, for example, 4 − 2 is just shorthand for 4 + (−2) and 4 / 2 is just shorthand for 4 · 2−1.

A parade of fields

We have now formally introduced the idea of an algebraic field. If we want to express intuitively what a field actually is, we can say that it is a structure that lets us add, subtract, multiply and divide any objects on which we can define these operations, in a way similar to how we carry out these operations in elementary arithmetic.

Our first field

Now that we have thought up and described fields so nicely, let us build a real one. First we choose the set of elements of the field. We will not experiment too much with cuddly elephants or socks for now, and will choose non-negative integers as the elements of the field. To keep everything really simple, we will take only the numbers 0, 1, 2, 3, 4. So our set T will look like this:

T={0,1,2,3,4}

All that remains is to define the operations ⊕ and ⊗ cleverly. We will make use of the addition and multiplication we already know from arithmetic, but modify them slightly so that they work on our field.

  • The operation ⊕ We find the sum of two elements a ⊕ b by working out the remainder when the sum a + b is divided by five. Addition in our field will therefore look like this: 1 ⊕ 1 = 2    1 ⊕ 3 = 4    2 ⊕ 3 = 0    4 ⊕ 2 = 1 and so on.
  • The operation ⊗ We find the product of two elements a ⊗ b by working out the remainder when the product ab is divided by five. Multiplication in our field will therefore look like this: 1 ⊗ 1 = 1    2 ⊗ 2 = 4    2 ⊗ 3 = 1    4 ⊗ 2 = 3    and so on.

And that is it. We now have a fully fledged field. We still need to check that all the axioms hold, to be sure that we have really made a field and not some monstrosity.

  • You can surely check the commutativity and associativity of addition and multiplication yourself, and the distributivity too.
  • From the way we defined addition and multiplication it also follows that there is a zero element (in our case it happens to be the number 0) and an identity element (in our case 1), and that they are different from each other.
  • So all that remains is to check the existence of the negative and inverse elements. We can all see that for every number in the set {0, 1, 2, 3, 4} there is a number in the same set such that the result of adding them is zero or five (that is, 0 in our field).
  • The same holds for multiplication: for every non-zero number in this set there is a number with which the product leaves remainder 1 when divided by five (that is, 1 in our field). For example, the number 4 has inverse element 4, because 4 · 4 = 16 and when we divide by 5 we get remainder 1. Try to find the other inverse elements!

Examples of other fields

A moment ago we built a field that has five elements. It turns out that in the same way we can build other fields where the number of elements is a prime number. Such fields are called fields of integers modulo p and are written Zp, where p is the number of elements of the field. They are very often used in maths and computer science, and of course in all sorts of exam questions too.

Can we make fields with a number of elements that is not prime? Yes we can, but not in the way I have just shown. Why? Imagine a field built by the procedure described above that has only four elements. In such a field 2 ⊗ 2 = 0. So the product of two non-zero numbers is zero. That must not happen in a field. It is not directly forbidden by the axioms of a field, but it can be derived from them quite easily. It turns out, however, that if we use polynomials instead of numbers as the elements of the field, we can build fields whose number of elements is a power of a prime. So a field with four elements does exist, since four is a power of two.

Of course, fields with infinitely many elements are possible too. For example, the set of real numbers together with our usual addition and multiplication also forms a field. This holds not only for the real numbers but also for the rational and complex numbers. The positive integers and the integers do not form a field, because they lack negative and inverse elements respectively.

And now the best part. Nobody says, of course, that fields have to be made up of numbers, polynomials or any mathematical objects at all. As I said, you can make a field of socks, cuddly elephants and so on. The only problem will be how to define addition and multiplication on elephants. What is interesting here is that you can never succeed in making a field that contains six socks. That is because six is neither a prime nor a power of a prime. But you can quite happily make a field that has seventeen socks, number each of them (the white sock will be 0, the black one 1, the green one 2 and so on) and then add them in the same way as we added in the field of integers modulo a prime.

And what is it all for?

If you have read this far, you may be asking yourself what on earth we need such a crazy structure as an algebraic field for. Well, fields are very important, both from the theoretical point of view, because they describe and above all generalise what could be called calculating, that is, arithmetic, and from the practical point of view, because many problems in maths or computer science are easier to solve if we imagine them as calculations in fields. Finite fields, for example, are very important for the codes on CDs and DVDs. As for linear algebra itself, fields are the cornerstone of other, more complex and therefore far more useful and interesting structures, such as vector spaces. These form the basis of the whole of linear algebra.

This article was written by Lishaak.