What is the field with one element ?

[Kamataat]

What is the "field with one element"?

From the definition of a field, it follows that such a thing does not exist. However a Google search reveals that apparently there is, or at least mathematicians think there ought to be, something that goes by that name. What is it? Is it called a "field" simply because of some analogies with fields even though it really isn't one?

The material (algebraic geometry) I've found online is over my head for the time being. I know basic abstract algebra, e.g. what a group, module, etc. is.

Also, the lecture notes I got from my uni mention the trivial field K = {a} with a*a=a, a+a=a and a=1=0.

Thanks in advance,
Kamataat

 

[Moo Of Doom]

This depends on your definition of field. My book says, "a field is a commutative ring with unity in which every nonzero element is a unit." But, my book defines a unity as a nonzero element of a ring that is a multiplicative identity. Therefore, there is no trivial field under this definition. If you simply throw out the condition that $1 \neq 0$, you can, of course define the trivial field.

 

[HallsofIvy]

From the definition of a field, it follows that such a thing does not exist. However a Google search reveals that apparently there is, or at least mathematicians think there ought to be, something that goes by that name. What is it? Is it called a "field" simply because of some analogies with fields even though it really isn't one?

The material (algebraic geometry) I've found online is over my head for the time being. I know basic abstract algebra, e.g. what a group, module, etc. is.

Also, the lecture notes I got from my uni mention the trivial field K = {a} with a*a=a, a+a=a and a=1=0.

Thanks in advance,
Kamataat

What Moo of Doom said is correct: most definitions of "field" require that there exist distinct 0 and 1. Dropping that requirement, then there can exist a "field" having only one element. It would have to be, then exactly what you give a "the trivial field".

 

[StatusX]

There is a ring with one element. It is commutative, has no zero divisors, has no proper ideals, etc. Whether you call this a field or not depends on whether you allow fields with 1=0, since it trivially satisfies every other property. And if you do, then for any element a in such a field a=a*1=a*0=0, so the field has exactly one element (which is why the condition that 1 and 0 be distinct is usually added).