A field, in abstract algebra, is a system where:
- Addition works the way you expect.
- Multiplication also works the way you expect.
- Multiplication distriutes over addition
We can make this more precise.
A field obeys the following axioms:
A1: Addition is commutative: for all x and y, x + y = y + x
A2: Addition is associative: for all x and y, (x + y) + z = x + (y + z)
A3: The field contains an additive identity: there exists a member 0 such that for all x, x + 0 = x
A4: Each member has an additive inverse: for each x there is a -x such that x + -x = 0
M1: Multiplication is commutative: for all x and y, x * y = y * x
M2: Multiplication is associative: for all x and y, (x * y) * z = x * (y * z)
M3: The field contains an multiplicative identity not equal to the additive identity: there exists a member 1 ≠ 0 such that for all x, x * 1 = x
M4: Each member other than the additive inverse has an multiplicative inverse: for each x ≠ 0 there is an x-1 such that x * x-1 = 1
D1: Multiplication distributes over addition: for all x, x, and z, x * (y + z) = (x * y) + (x * z)
That’s the whole thing. The most familiar fields are the rational numbers, real number, and complex numbers . There is also a collection of finite fields. For each prime number P, the field consists of the numbers 0 through P-1. Addition and multiplication are normal, except that if the answer would be P or greater, you replace it with the remained you’d get dividing it by P. This is called modulo arithmetic. For instance, in the field with members 0, 1, 2, 3, and 4, 2 * 3 = 1, because 6 divided by 5 leaves the remainder 1. Here are the addition and multiplication tables for this field. You can verify all the axioms, if you have the patience.
Lemma: For all x, x * 0 = 0
We know that 0 + 1 = 0, so x = x * 1 = x * (1 + 0)
Using D1, x = (x * 1) + (x * 0) = x + (x * 0)
Substituting equals for equals into 0 = -x + x,
we get 0 = -x + x + (x * 0) = 0 + (x * 0) = x * 0
Using the lemma, if 0 = 1 then for all x, x * 1 = 0, or x = 0. So all numbers are equal to zero, and thus to each other. The result is a very complete system in which there is one answer to almost all questions. Is there a square root of two? Yes, it’s 0. Is there a square root of -1? Yup, 0. Can you divide by 0? Sure, and the result is 0.
Any resemblance to your least favorite ideology, which considers every issue a nail to be attacked with the same sledgehammer, is purely coincidental.
1. There’s a four-dimensional analog of the complex numbers called the quaternions, where in addition to 1 and i you have j and k, but it’s not quite a field because multiplication isn’t commutative. i * j = k, but j * i = -k