if F is a field show that Fx1 xn is an integral domainSolut

if F is a field, show that F[x1, ..., xn] is an integral domain.

Solution

If F is a field, then  it is a commutative ring with identity (1 0) in which every non-zero element has a multiplicative inverse. Again an integral domain is a commutative ring with an identity (1 0) with no zero-divisors.
That is ab = 0 a = 0 or b = 0.

Here F is a field it is a commutative ring with identity and inverse

It means if a, b F then a*b = b*a

there exists an identity I F such that a*I = I*a

and for inverse, there exists a & e such that a * a\' = e = a\'*a

Now F is also an Integral Domain as for Two non-zero matrices of order 2*2 is zero and as such it is an integral domain. For example

A = ( 0 1 ), B = ( 1 0 ) , but AB = ( 0 0 ), so A 0, B 0, but AB = 0.

0 0 0 0 0 0