Question

If p (x)is a nonzero constant polynomial in F [x] , show that any two polynomials in F [x] are congruent modulo p (x).

Short answer

It is proved that two polynomials inF [x] are congruent modulo p (x).

Step-by-step solution

Given part

It is given that p (x) is a non-zero polynomial in F [x].

Let the constant be c.

Again, assume that, f (x) and g (x)are arbitrary polynomials.

Proof part

It is given that F [x] is a field, so the inverse of constant c is c^-1.

Then $c^{-1}\left(f\left(x\right)-g\left(x\right)\right)\in F\left[x\right]$.

Which implies $c|c^{-1}c\left(f\left(x\right)-g\left(x\right)\right)=f\left(x\right)-g\left(x\right)$.

Therefore, $p\left(x\right)|\left(f\left(x\right)-g\left(x\right)\right)$.

It can be said that any two polynomials in F [x] are congruent modulo p (x).

Hence, we get $f\left(x\right)\equiv g\left(x\right)\mathrm{mod}p\left(x\right)$.