If F is a field, show that is not a field.
It is proved that F[x] is not a field.
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.
It is given that F is a field.
Let us assume that F[x] is also a field. Then, will have an inverse as:
Therefore, in this case, we have:
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.
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