Question

Show that every field is an integral domain

Answer 1

Answer:

Theorem 2:

A field is necessarily an integral domain.

Step-by-step explanation:

Proof:

Since a field is a commutative ring with unity, therefore, in order to show that every field is an integral domain we only need to prove that s field is without zero divisors.

Let F be any field and let a,b∈F with a≠0 such that ab=0. Let 1 be the unity of F. Since a≠0, a−1 exists in F, therefore

ab=0⇒a−1(ab)=a−10⇒(a−1a)b=0⇒1⋅b=0⇒b=0

Similarly if b≠0 then it can be shown that ab=0⇒a=0.

Thus ab=0⇒a=0orb=0. Hence, a field is necessarily an integral domain.