Question

If F is a field, show that $F\left[x\right]$ is not a field.

Short answer

It is proved that F[x] is not a field.

Step-by-step solution

Polynomial Arithmetic

If any given function R[x] is a ring, then the commutative, associative, and distributive laws hold such that the function f(x)+g(x) exists.

Fields:

It is given that F is a field.

Let us assume that F[x] is also a field. Then, $\forall x\in F\left[x\right]$ will have an inverse as:

$f\left(x\right)=\sum _{i=0}^m{a}_i{x}^i$

Therefore, in this case, we have:

$xf\left(x\right)=\sum _{i=0}^m{a}_i{x}^{i+1}=1$

Here, the constant coefficient is zero.

According to the theorem 4.1, it should be 1.

This shows our assumption is invalid.

Hence proved, F[x] is not a field.