# Prove that a finite subring of a field is a field.

Prove that a finite subring R of a field F is itself a field. Hint: if x is an element of R and x is not equal to 0 show the function f:R->R with f(r) = xr is injective. From finiteness of R, deduce that its image includes 1

© BrainMass Inc. brainmass.com June 4, 2020, 12:42 am ad1c9bdddfhttps://brainmass.com/math/ring-theory/finite-subring-field-354415

#### Solution Preview

Let x be in R, and x is not equal to 0.

Then, the function f: R->R defined by f(r) = xr is injective.

First, show that f is well-defined. If r is in R and x is in R, then xr is in R, since R is a ring.

Now, show that f is injective. Suppose, there are two elements r1 and r2 in R such that f(r1) = f(r2).

But f(r1) = xr1, f(r2)=xr2, and so

xr1 = xr2

Since x is a non-zero element of a ...

#### Solution Summary

We give a rigorous proof that a finite subring of a field is a field itself.

$2.19