# Field of Fractions

1. Sep 8, 2010

### beetle2

Hi guys,

I know that for integral domains with finte elements that if we show that each element has a multiplicative inverse then it is a field.

I need to show that the field of fractions is a field.

As the domain is not finite how does that effect the proof of being a field?

regards
Brendan

2. Sep 8, 2010

### Office_Shredder

Staff Emeritus
Whether the domain is finite or not will be irrelevant unless you have a pretty wacky proof.

Your proof should probably be inspired by how you would prove that the rational numbers are a field, which is obviously infinite

3. Sep 9, 2010

### Landau

The definition of a field is an integral domain in which each non-zero element has an inverse. I don't see where you get the finiteness condition from.
So just take an arbitrary non-zero element in the field of fractions and show that it has an inverse.

4. Sep 12, 2010

Is this what you meant by the first question? "If an integral domain is finite then it is a field." This is fairly well-known. To prove it: Suppose x is a nonzero element of a finite integral domain; then find positive integers k > l such that xk = xl. Then prove that xxk-l-1 = 1.

5. Sep 12, 2010

### Skolem

If adriank is correct, and you mean prove A. "If an integral domain is finite then it is a field." and B. "Why does this fail if the integral domain is not finite?" then consider the following:

First, if you have an integral domain D any a (not 0) inside D then the function
mult_a: D --> D given by
mult_a(x) = a*x
is injective -exactly- because D is an integral domain:
mult_a(x) = mult_a(y) => a*x = a*y => x=y by cancellation property of integral domains.

But if is D finite, mult_a is also surjective, so for some x, mult_a(x) = 1. So what does this mean about x?

Second, if D is not finite then you can't conclude anything: the function f: Z --> Z (Z = integers) given by f(x) = 2*x is injective but certainly not surjective, and Z is an integral domain.

Anyways, that an injective function f: A --> A is surjective if A is finite is a very important fact in mathematics. That this can fail if A is infinite is equally important. You can use this last fact to construct a (nontrivial) group G so that G x G is isomorphic to G, for example.

Skolem