Prove that every finite domain D ie D SolutionEvery field F

Prove that every finite domain D (i.e., |D|

Every field F is an integral domain since ab = 0 and a 0 imply that b = 1b = (a ^-1 a)b = a^-1 (ab) = a ^-1 0 = 0.

Let D be a finite integral domain whose unity is 1 and whose zero is 0. Let a D such that a 0.

We will show that a has a product inverse in D. So consider the map f: DD defined by f: xax.

We first show that the kernel of f is trivial.Let us consider that:

ker(f)={xD: f(x)=0}={x D: ax=0}

Since D is an integral domain, it has no proper zero divisors and thus ax = 0 means that either a =0 or x=0.

Since, by definition, a0, then it must be true that x=0.

Therefore, ker (f)={0} and so f is injective.

Next, as per the Pigeonhole Principle, an injective mapping from a finite set onto itself is surjective.

Since D is finite, the mapping f is surjective.

Finally, since f is surjective and 1D, there exists x D:f(x) = ax = 1

So this x is the product inverse of a. Thus, the proof is complete.